Finite Linn er Set Theory Finite MathematicsFinite Mathematics Linn er Set Theory Finite sets. Set...

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Universal setand empty set.

Subsets.

Sets of sets.

Set-operations.

Examples.

Commutativeand associative.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Multiplicationprinciple.

Countingsubsets.

Finite Mathematics

Anders Linneralinner@math.niu.edu

It is all about methodologies!

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

Starting Point

1 Set TheoryFinite sets.Set notation.Universal set and empty set.Subsets.Sets of sets.Set-operations.Examples.Laws.Commutative and associative.DeMorgan’s laws.Proof I.Proof II.Venn-diagram I.Venn-diagram II.Exclusive OR.Counting.Counting subsets.Notation

2 CountingMultiplication principle.Counting subsets.

It is difficult to define what a set is in general.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

Starting Point

1 Set TheoryFinite sets.Set notation.Universal set and empty set.Subsets.Sets of sets.Set-operations.Examples.Laws.Commutative and associative.DeMorgan’s laws.Proof I.Proof II.Venn-diagram I.Venn-diagram II.Exclusive OR.Counting.Counting subsets.Notation

2 CountingMultiplication principle.Counting subsets.

It is difficult to define what a set is in general.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

The simplest case.Consider only a finite number of elements.

Start things by specifying all available elements.

Use list notation.

U = {a, b, c , d , e, f , g} .

The order of the listed elements is not important.

There should not be any duplicates.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

Use list notation.

U = {a, b, c , d , e, f , g} .

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

Use list notation.

U = {a, b, c , d , e, f , g} .

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

Use list notation.

U = {a, b, c , d , e, f , g} .

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

Use list notation.

U = {a, b, c , d , e, f , g} .

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

Once U is specified, a set is simply a sample of the availableelements.

Examples include{b, c , f , g}

and{a, e, f } .

It is possible to choose all available elements, and thecorresponding set U is known as the universal set.

It is also possible to choose none of the available elements, andthe corresponding set is known as the empty set.

The empty set is denoted by ∅, and it has no elements.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

and{a, e, f } .

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

and{a, e, f } .

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

and{a, e, f } .

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

and{a, e, f } .

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

and{a, e, f } .

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

If each element in a set A is also an element of a set B, thenwrite A ⊂ B.

Observe that ∅ ⊂ A, A ⊂ A, and A ⊂ U no matter what the setA is.

Write A = B whenever A ⊂ B and B ⊂ A.

The subsets of {a, b, c} are

{∅, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}} .

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

{∅, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}} .

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

{∅, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}} .

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

{∅, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}} .

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

The previous listing of subsets looks suspiciously similar to aset.

Since it seems reasonable to in fact consider it a set, this vastlybroadens the scope of set theory.

What is the difference between ∅ and {∅}?

The former has no elements whereas the latter has oneelement, the empty set.

The set of all subsets of a set is known as the power set of thegiven set.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

There are three important ways to create sets from other sets.

Union, ∪, OR.

Intersection, ∩, AND.

Complement, Ac , NOT.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

Union, ∪, OR.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

Union, ∪, OR.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

Union, ∪, OR.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

Let U = {a, b, c , d , e, f , g , h, i , j}, A = {b, c , g , i , j}, andB = {a, b, c , d , e, j}.

Then A ∪ B = {a, b, c , d , e, g , i , j}, A ∩ B = {b, c , j},Ac = {a, d , e, f , h}, and Bc = {f , g , h, i}.

Observe that a universal set must be given in order todetermine the complement.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

There are a number of set-laws worth knowing.

∅c = UUc = ∅(Ac)c = AB ∪ A = A ∪ BA ∪ (B ∪ C ) =(A ∪ B) ∪ C

A ∪ A = AA ∩ A = AA ∪ ∅ = AB ∩ A = A ∩ BA ∩ (B ∩ C ) =(A ∩ B) ∩ C

A ∩ ∅ = ∅A ∪ U = UA ∩ U = A

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

There are a number of set-laws worth knowing.

∅c = UUc = ∅(Ac)c = AB ∪ A = A ∪ BA ∪ (B ∪ C ) =(A ∪ B) ∪ C

A ∪ A = AA ∩ A = AA ∪ ∅ = AB ∩ A = A ∩ BA ∩ (B ∩ C ) =(A ∩ B) ∩ C

A ∩ ∅ = ∅A ∪ U = UA ∩ U = A

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

The terminology ‘both unions and intersections arecommutative as well as associative’ is supported by these laws.

The listed laws do not mix operations.

What about the distributive law?

There are two distributive laws.

A ∪ (B ∩ C ) = (A ∪ B) ∩ (A ∪ C )

A ∩ (B ∪ C ) = (A ∩ B) ∪ (A ∩ C )

The next two laws have no counterpart in arithmetic.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

A ∪ (B ∩ C ) = (A ∪ B) ∩ (A ∪ C )

A ∩ (B ∪ C ) = (A ∩ B) ∪ (A ∩ C )

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

A ∪ (B ∩ C ) = (A ∪ B) ∩ (A ∪ C )

A ∩ (B ∪ C ) = (A ∩ B) ∪ (A ∩ C )

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

A ∪ (B ∩ C ) = (A ∪ B) ∩ (A ∪ C )

A ∩ (B ∪ C ) = (A ∩ B) ∪ (A ∩ C )

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

A ∪ (B ∩ C ) = (A ∪ B) ∩ (A ∪ C )

A ∩ (B ∪ C ) = (A ∩ B) ∪ (A ∩ C )

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

A ∪ (B ∩ C ) = (A ∪ B) ∩ (A ∪ C )

A ∩ (B ∪ C ) = (A ∩ B) ∪ (A ∩ C )

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

A ∪ (B ∩ C ) = (A ∪ B) ∩ (A ∪ C )

A ∩ (B ∪ C ) = (A ∩ B) ∪ (A ∩ C )

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

The following is true

(A ∪ B)c = Ac ∩ Bc .

If the claim is correct, then applying the complement to bothsides produces A ∪ B = (Ac ∩ Bc)c .

Replace A by Ac and B by Bc , a mere change in namingconvention, and rearrange to get

(A ∩ B)c = Ac ∪ Bc .

These are known as deMorgan’s laws.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(A ∪ B)c = Ac ∩ Bc .

(A ∩ B)c = Ac ∪ Bc .

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(A ∪ B)c = Ac ∩ Bc .

(A ∩ B)c = Ac ∪ Bc .

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(A ∪ B)c = Ac ∩ Bc .

(A ∩ B)c = Ac ∪ Bc .

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

It is of course possible to test this claim in some simpleexamples, but if this is to be true no matter which sets areinvolved some other reasoning must be employed.

To break things down into smaller steps, first consider theclaim that (A ∩ B)c ⊂ Ac ∪ Bc .

Suppose x ∈ (A ∩ B)c .

It follows that x /∈ A ∩ B.

It may be that x is in one or the other of the two sets but it isnot in both.

If x is not in A then x ∈ Ac and x ∈ Ac ∪ Bc .

Similarly, if x is not in B then x ∈ Bc and x ∈ Ac ∪ Bc .

In either case x ∈ Ac ∪ Bc and hence (A ∩ B)c ⊂ Ac ∪ Bc .

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

Now turn the attention to (A ∩ B)c ⊃ Ac ∪ Bc .

This is just different notation for Ac ∪ Bc ⊂ (A ∩ B)c .

This time suppose x /∈ (A ∩ B)c .

This means x is in A ∩ B.

It follows that x ∈ A and x ∈ B.

This means x /∈ Ac and x /∈ Bc .

So it follows that x /∈ Ac ∪ Bc and (A ∩ B)c ⊃ Ac ∪ Bc .

The two set inclusions combine to guarantee that(A ∩ B)c = Ac ∪ Bc .

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

A so-called Venn-diagram may be used to establish less familiarset-identities.

The simplest Venn-diagram consists of a rectangle thatrepresents the universal set of interest, and two ovals thatoverlap and represent A and B, respectively.

The universal set splits into four parts: A ∩ B, A ∩ Bc , Ac ∩ B,and Ac ∩ Bc .

According to deMorgan, the last set is everything outsideA ∪ B.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

The second kind of Venn-diagram has three ovals correspondingto A, B, and C that are drawn in the the most general position.

This time the universal set splits into eight parts:

A ∩ B ∩ C A ∩ B ∩ C c

A ∩ Bc ∩ CAc ∩ B ∩ C

A ∩ Bc ∩ C c

Ac ∩ B ∩ C c

Ac ∩ Bc ∩ C

Ac ∩ Bc ∩ C c

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

The second kind of Venn-diagram has three ovals correspondingto A, B, and C that are drawn in the the most general position.

This time the universal set splits into eight parts:

A ∩ B ∩ C A ∩ B ∩ C c

A ∩ Bc ∩ CAc ∩ B ∩ C

A ∩ Bc ∩ C c

Ac ∩ B ∩ C c

Ac ∩ Bc ∩ C

Ac ∩ Bc ∩ C c

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

A subset set created from A, B and C using set-operationsinvolves the inclusion or exclusion of each of the eight parts sothere are 28 = 256 possibilities.

The Venn-diagram will in each case resolve things as onesimply writes a union of the parts involved.

The so-called ‘exclusive or’, which involves exactly the twoparts A∩Bc and Ac ∩B, is represented by (A∩Bc)∪ (Ac ∩B).

In plain language use, ‘one or the other’ often means exclusiveor.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

Write n(A) for the number of elements in the set A.

Observe that n(∅) = 0.

The generally true

n(A ∪ B) = n(A) + n(B)− n(A ∩ B)

is connected with the phrase ‘inclusion-exclusion’.

A and B are said to be mutually exclusive if A ∩ B = ∅, and inthis case n(A ∪ B) = n(A) + n(B).

It also follows that n(Ac) = n(U)− n(A).

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

The generally true

n(A ∪ B) = n(A) + n(B)− n(A ∩ B)

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

The generally true

n(A ∪ B) = n(A) + n(B)− n(A ∩ B)

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

The generally true

n(A ∪ B) = n(A) + n(B)− n(A ∩ B)

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

The generally true

n(A ∪ B) = n(A) + n(B)− n(A ∩ B)

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

Theorem

The number of subsets of a finite set A is

2n(A).

Proof.

1 Suppose the number of subsets of a set is x .

2 Now add one extra element to the set.

3 All the previous subsets continue to be subsets.

4 By adding the extra element to each of the previous subsets,another distinct x sets are created for a total of 2x .

5 The empty set has 0 elements and 1 subset, itself.

6 By doubling, this creates the sequence, 20, 21, 22, and soon and so forth.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

Theorem

2n(A).

Proof.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

Theorem

2n(A).

Proof.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

Theorem

2n(A).

Proof.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

Theorem

2n(A).

Proof.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

Theorem

2n(A).

Proof.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

The listing of all elements is only practical when the number ofelements is fairly small.

There is an alternative notation available.

{x | 1 ≤ x ≤ 99, x even integer}.

One also writes x ∈ Z to indicate that x is an integer.

In this way infinite sets may be described.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

True False Tests

How many ways are there to fill out a 2 question True–FalseExam?

Since there are 2 ways to answer the first question and 2 waysto answer the second question, there are 2× 2 = 4 possibleanswer keys:

TT TF FT FF

How many ways are there to fill out a 3 question True–FalseExam?Since there are 2 ways to answer each of the three questions,there are 23 = 8 possible answer keys:

TTT TTF TFT TFF FTT FTF FFT FFF

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

True False Tests

How many ways are there to fill out a 2 question True–FalseExam?Since there are 2 ways to answer the first question and 2 waysto answer the second question, there are 2× 2 = 4 possibleanswer keys:

TT TF FT FF

How many ways are there to fill out a 3 question True–FalseExam?

Since there are 2 ways to answer each of the three questions,there are 23 = 8 possible answer keys:

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

True False Tests

How many ways are there to fill out a 2 question True–FalseExam?Since there are 2 ways to answer the first question and 2 waysto answer the second question, there are 2× 2 = 4 possibleanswer keys:

TT TF FT FF

How many ways are there to fill out a 3 question True–FalseExam?Since there are 2 ways to answer each of the three questions,there are 23 = 8 possible answer keys:

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

Given a finite set, a subset A is determined by indicating eachof its elements as true, and each other element as true.

The number of ways this can be done is analogous to theTrue-False test, so the count is 2n(A).

Consider a task that involve several intermediate steps thatmust be completed in order.

To be specific, suppose that all two-character codes of the form‘uppercase letter followed by digit’ are to be typed.

This task has two intermediate steps.

The multiplication principle asserts that to count the numberof ways the task may be completed simply multiply the numberof ways each intermediate step may be completed.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

In the example the first step may be completed in 26 ways, andthe second in 10 ways.

The number of ways the task may be completed is therefore26 · 10 = 260.

Observe that even for relatively small values the total count islarge enough that listing all the possibilities seems like thewrong approach.

To aid in these calculations there are several usefulmathematical functions.

When each object in a finite collection is given a uniquedesignation, then the count of the number of possibilitiesutilizes the so-called factorial.

Let the collection be {a, b, c}, and list the members in order.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

The selection of the first has three possibilities.

Once the first is selected, only two remain so only twopossibilities.

For the third there is no choice.

The multiplication principle gives 3 · 2 · 1 = 6 ways.

Write 3! for the product 3 · 2 · 1, and more generally 1! = 1together with n! = n · (n − 1)!.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

Here is a table for the factorial.

3 64 245 1206 7207 50408 403209 362880

10 362880011 3991680012 47900160013 622702080014 8717829120015 130767436800016 2092278988800017 355687428096000

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

Observe how quickly the values grow.

Calculators have a factorial key but may not give 14!accurately, let alone 70!.

There are many applications where only some of the availableobjects are to be given a designation.

In the 100m dash there are 8 runners competing for first,second, and third.

By the multiplication principle, the number of ways the podiummay be filled is 8 · 7 · 6 = 336.

Observe that this value is in fact 8!5! .

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

This suggests defining the permutation functionP(n, r) = n!

(n−r)! .

In order for this formula to work in complete generality it isnecessary to define 0! = 1, although this at first sight seemsfar-fetched.

Now it seen that P(n, n) = P(n, n − 1) = n!, P(n, 1) = n, andP(n, 0) = 1.

The permutation function may also be available on a calculatorand it suffers from limitations due to the size of the valuesproduced, just like the factorial.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(n−r)! .

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(n−r)! .

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(n−r)! .

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

There are many times when a selection is made withoutsubsequent designation.

If each available object is selected, then there is little to saysince the selection is viewed as a set and there is one possibleset.

Technically the calculation is P(n,n)n! = n!

0!·n! = 1.

The n! in the denominator accounts for all the ways the nobjects may be arranged in order.

If no object is selected, then again there is not much to say andthe formula gives P(n,0)

0! = n!n!·0! = 1.

‘There is only one way to select all objects or none of theobjects’.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

0!·n! = 1.

0! = n!n!·0! = 1.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

0!·n! = 1.

0! = n!n!·0! = 1.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

0!·n! = 1.

0! = n!n!·0! = 1.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

0!·n! = 1.

0! = n!n!·0! = 1.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

0!·n! = 1.

0! = n!n!·0! = 1.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

There are n ways to select exactly one object.

There are n ways to select all but one of the objects.

The formulas are

P(n, 1)

(n − 1)! · 1!= n,

andP(n, n − 1)

(n − 1)!=

1! · (n − 1)!= n.

Define the combination function C (n, r) by

C (n, r) =P(n, r)

(n − r)! · r !.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

The formulas are

P(n, 1)

(n − 1)! · 1!= n,

andP(n, n − 1)

(n − 1)!=

1! · (n − 1)!= n.

C (n, r) =P(n, r)

(n − r)! · r !.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

The formulas are

P(n, 1)

(n − 1)! · 1!= n,

andP(n, n − 1)

(n − 1)!=

1! · (n − 1)!= n.

C (n, r) =P(n, r)

(n − r)! · r !.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

The formulas are

P(n, 1)

(n − 1)! · 1!= n,

andP(n, n − 1)

(n − 1)!=

1! · (n − 1)!= n.

C (n, r) =P(n, r)

(n − r)! · r !.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

As seen, by thinking about the objects selected versus theobjects left behind it must be that C (n, r) = C (n, n − r).

To illustrate this, C (2015, 2013) would produce hopelessly largevalues in the formula.

C (2015, 2013) = C (2015, 2015− 2013) = C (2015, 2),

C (2015, 2) =P(2015, 2)

2015 · 2014

2= 2015 · 1007

2015 · 1007 = 2015(1000 + 7) = 2015000 + 14105 = 2029105.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

C (2015, 2013) = C (2015, 2015− 2013) = C (2015, 2),

C (2015, 2) =P(2015, 2)

2015 · 2014

2= 2015 · 1007

2015 · 1007 = 2015(1000 + 7) = 2015000 + 14105 = 2029105.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

C (2015, 2013) = C (2015, 2015− 2013) = C (2015, 2),

C (2015, 2) =P(2015, 2)

2015 · 2014

2= 2015 · 1007

2015 · 1007 = 2015(1000 + 7) = 2015000 + 14105 = 2029105.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

Think of splitting the available objects into two groups, theselected ones and the ones left behind.

By the multiplication principle, before adjusting to theindifference about the ordering, the count isP(n, r) · P(n − r , n − r) = n!

(n−r)! ·(n−r)!

0! = n!.

Now divide out the number of ways of arranging the r selectedobjects as well as the number of ways of arranging the objectsleft behind so n!

r !(n−r)! , which again is C (n, r).

This line of thinking works when the objects split into kgroups, each containing ni members so that n1 + · · ·+ nk = n.

In this case, with the k designations in order, there are n!n1!···nk !

possibilities.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(n−r)! ·(n−r)!

0! = n!.

possibilities.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(n−r)! ·(n−r)!

0! = n!.

possibilities.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(n−r)! ·(n−r)!

0! = n!.

possibilities.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(n−r)! ·(n−r)!

0! = n!.

possibilities.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

Consider the objects in S = {a1, a2, a3, b1, b2, c}.

There are 6 distinguishable objects.

Now consider the collection a, a, a, b, b, c .

This time there are only three distinguishable items: a, b, andc .

The number of permutations of the objects in S is 6! = 720.

One sees the difference between a1a2a3b1b2c and a2a1a3b1b2c .

If the subscripts are removed then the difference disappear asboth look like aaabbc.

The question now is how many permutations there are whensome of the items are indistinguishable.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

To answer that question it suffices to divide by the number ofpermutations of each collection of indistinguishable objects.

In the case of S and a, a, a, b, b, c it follows the count is

3!2!1!=

6 · 5 · 4 · 3 · 23 · 2 · 2

It is important to remember that the order is important soaaabbc is different than aababc.

To analyze the formula observe that the c can be placed inanyone of the six positions.

Now make those placements in each of the following 10 cases:aaabb, aabab, aabba, abaab, ababa, abbaa, baaab, baaba,babaa, and bbaaa.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

3!2!1!=

6 · 5 · 4 · 3 · 23 · 2 · 2

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

3!2!1!=

6 · 5 · 4 · 3 · 23 · 2 · 2

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

3!2!1!=

6 · 5 · 4 · 3 · 23 · 2 · 2

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

3!2!1!=

6 · 5 · 4 · 3 · 23 · 2 · 2

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

This produces the 60 possibilities.

The expression n!n1!···nk ! is known as a multinomial coefficient.

In the special case when k = 2 it is known as the binomialcoefficient.

Recall that (x + y)2 = x2 + 2xy + y2 and it follows that

(x + y)3 = (x + y)(x2 + 2xy + y2) = x3 + 3x2y + 3xy2 + y3.

By thinking about how (x + y)(x + y)(x + y) produces thevarious terms in the expansion one realizes that the binomialcoefficient is involved.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(x + y)3 = (x + y)(x2 + 2xy + y2) = x3 + 3x2y + 3xy2 + y3.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(x + y)3 = (x + y)(x2 + 2xy + y2) = x3 + 3x2y + 3xy2 + y3.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(x + y)3 = (x + y)(x2 + 2xy + y2) = x3 + 3x2y + 3xy2 + y3.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(x + y)3 = (x + y)(x2 + 2xy + y2) = x3 + 3x2y + 3xy2 + y3.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

Observe that C (2, 0) = 1, C (2, 1) = 2, and C (2, 2) = 1.

Similarly, C (3, 0) = 1, C (3, 1) = 3, C (3, 2), and C (3, 3) = 1.

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

It follows that (x + y)4 = x4 + 4x3y + 6x2y2 + 4xy3 + y4.

Also, (x + y)5 = x5 + 5x4y + 10x3y2 + 10x2y3 + 5xy4 + y5.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

More generally,(x + y + z)3 =

x3 + 3x2y + 3x2z + 3xy2 + 6xyz + 3xz2 + y3 + 3y2z + 3yz2 + z3.

This time 3!3!0!0! = 1, 3!

2!1!0! = 3, and 3!1!1!1! = 6, which are seen

as coefficients in the expansion.

In the binomial case set x = y = 1 so that(x + y)n = (1 + 1)n = 2n.

This implies that

C (n, 0) + C (n, 1) + · · ·+ C (n, n − 1) + C (n, n) = 2n.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

This time 3!3!0!0! = 1, 3!

This implies that

C (n, 0) + C (n, 1) + · · ·+ C (n, n − 1) + C (n, n) = 2n.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

This time 3!3!0!0! = 1, 3!

This implies that

C (n, 0) + C (n, 1) + · · ·+ C (n, n − 1) + C (n, n) = 2n.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

This time 3!3!0!0! = 1, 3!

This implies that

C (n, 0) + C (n, 1) + · · ·+ C (n, n − 1) + C (n, n) = 2n.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

When counting subsets of a set with n elements, the number ofsubsets with k elements is given by C (n, k).

This then connects with the fact that the total number ofsubsets of a set with n elements is 2n.

A different point of view is provided by the idea of tagging eachelement by its own bit.

Set the bit to 1 if the element is part of the subset and 0 if it isnot.

This creates a correspondence between subsets of sets with nelements and binary numbers with n digits.

In particular this shows that there are 2n different binarynumbers with n digits.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

The first one 00 · · · 00 corresponds to the empty set, whereasthe last one 11 · · · 11 corresponds to the whole set.

Inside computers there is a translation so that 00 · · · 00corresponds to the number 0, whereas 11 · · · 11 corresponds tothe number 2n − 1.

In the common case when n = 8 this means 00000000corresponds to 0, whereas 11111111 corresponds to28 − 1 = 256− 1 = 255.

So-called hex-code where 0000→ 0, 0001→ 1, ..., 0111→ 7,1000→ 8, 1001→ 9, 1010→ A, 1011→ B, 1100→ C ,1101→ D, 1110→ E , and 1111→ F , implies that 00corresponds to 0 and FF corresponds to 255.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

Which of the following is true ifA = {1, 2, 4, 6, 9}, B = {1, 5, 7, 9}, U = {1, 2, 3, 4, 5, 6, 7, 8, 9}?

(a) n(A ∪ B) = 9

(b) {1} ∈ A

(c) A ∩ Bc = {5, 7}

(d) A and B are mutually exclusive.

(e) {1, 9} = A ∩ B

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) n(A ∪ B) = 9

(b) {1} ∈ A

(c) A ∩ Bc = {5, 7}

(e) {1, 9} = A ∩ B

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) n(A ∪ B) = 9

(b) {1} ∈ A

(c) A ∩ Bc = {5, 7}

(e) {1, 9} = A ∩ B

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) n(A ∪ B) = 9

(b) {1} ∈ A

(c) A ∩ Bc = {5, 7}

(e) {1, 9} = A ∩ B

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) n(A ∪ B) = 9

(b) {1} ∈ A

(c) A ∩ Bc = {5, 7}

(e) {1, 9} = A ∩ B

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) n(A ∪ B) = 9

(b) {1} ∈ A

(c) A ∩ Bc = {5, 7}

(e) {1, 9} = A ∩ B

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

A ∪ B = {1, 2, 3, 4, 5, 6, 9} so n(A ∪ B) = 7.

Observe that 1 ∈ A but not {1} ∈ A.

Bc = {2, 3, 4, 6, 8} so A ∩ Bc = {2, 4}.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

A ∪ B = {1, 2, 3, 4, 5, 6, 9} so n(A ∪ B) = 7.

Bc = {2, 3, 4, 6, 8} so A ∩ Bc = {2, 4}.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

A ∪ B = {1, 2, 3, 4, 5, 6, 9} so n(A ∪ B) = 7.

Bc = {2, 3, 4, 6, 8} so A ∩ Bc = {2, 4}.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

Let U = {1, 2, 3, 4, 5, 6, a, b, c, d , e} and A = {1, 2, a, e} thenn(Ac) =

(c) 12

(d) 10

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(c) 12

(d) 10

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(c) 12

(d) 10

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(c) 12

(d) 10

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(c) 12

(d) 10

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(c) 12

(d) 10

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

Let S = {a, e, l , t, r} and consider:

A = {x |x is a letter in late},

B = {x |x is a letter in elate},

C = {x |x is a letter in latter}.

Which statement is true?

(a) S ,A,B,C are distinct sets.

(b) S = A, but is not the same as B or C .

(c) S = B, but is not the same as A or C .

(d) S = A = B = C

(e) S = C , but is not the same as A or B.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(d) S = A = B = C

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(d) S = A = B = C

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(d) S = A = B = C

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(d) S = A = B = C

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(d) S = A = B = C

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

A book club has a selection of 10 books one month. Howmany ways are there of picking two books?

(a) 90

(b) 50

(d) 100

(e) 45

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) 90

(b) 50

(d) 100

(e) 45

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) 90

(b) 50

(d) 100

(e) 45

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) 90

(b) 50

(d) 100

(e) 45

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) 90

(b) 50

(d) 100

(e) 45

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) 90

(b) 50

(d) 100

(e) 45

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

Five houses in a row are to be painted either Red, Blue, Greenor Yellow. In how many ways can this be done if adjacenthouses are to be painted different colors?

(a) 5 · 4 · 3 · 2

(b) 4 · 3 · 3 · 3 · 3

(c) 5 · 4 · 4 · 4

(d) 3 + 2 + 1 + 0

(e) None of the above.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) 5 · 4 · 3 · 2

(b) 4 · 3 · 3 · 3 · 3

(c) 5 · 4 · 4 · 4

(d) 3 + 2 + 1 + 0

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) 5 · 4 · 3 · 2

(b) 4 · 3 · 3 · 3 · 3

(c) 5 · 4 · 4 · 4

(d) 3 + 2 + 1 + 0

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) 5 · 4 · 3 · 2

(b) 4 · 3 · 3 · 3 · 3

(c) 5 · 4 · 4 · 4

(d) 3 + 2 + 1 + 0

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) 5 · 4 · 3 · 2

(b) 4 · 3 · 3 · 3 · 3

(c) 5 · 4 · 4 · 4

(d) 3 + 2 + 1 + 0

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) 5 · 4 · 3 · 2

(b) 4 · 3 · 3 · 3 · 3

(c) 5 · 4 · 4 · 4

(d) 3 + 2 + 1 + 0

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

How many six-digit telephone numbers are possible if the digitsare not all zero?

(a) 999, 999

(b) 900, 000

(c) 700, 000

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) 999, 999

(b) 900, 000

(c) 700, 000

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) 999, 999

(b) 900, 000

(c) 700, 000

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) 999, 999

(b) 900, 000

(c) 700, 000

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) 999, 999

(b) 900, 000

(c) 700, 000

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) 999, 999

(b) 900, 000

(c) 700, 000

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

What is the value represented by C (5, 2).

(a) 60

(b) 20

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) 60

(b) 20

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) 60

(b) 20

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) 60

(b) 20

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) 60

(b) 20

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) 60

(b) 20

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

From a department of 15 people, how many ways are there ofchoosing a committee consisting of a president, a vicepresident, and a secretary?

(a) 2730

(b) 455

(c) 3375

(d) 563

(e) None of these.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) 2730

(b) 455

(c) 3375

(d) 563

(e) None of these.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) 2730

(b) 455

(c) 3375

(d) 563

(e) None of these.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) 2730

(b) 455

(c) 3375

(d) 563

(e) None of these.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) 2730

(b) 455

(c) 3375

(d) 563

(e) None of these.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) 2730

(b) 455

(c) 3375

(d) 563

(e) None of these.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

A child’s toy consists of a head, torso, legs and arms. If thereare 5 choices of head, 7 for torso, 3 for legs, and 6 for arms,how many different toys are possible?

(a) 105

(b) 126

(c) 450

(d) 630

(e) None of these.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) 105

(b) 126

(c) 450

(d) 630

(e) None of these.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) 105

(b) 126

(c) 450

(d) 630

(e) None of these.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) 105

(b) 126

(c) 450

(d) 630

(e) None of these.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) 105

(b) 126

(c) 450

(d) 630

(e) None of these.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) 105

(b) 126

(c) 450

(d) 630

(e) None of these.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

In a class of 100 students, 10 jog, 25 swim and 5 both jog andswim. How many neither jog nor swim?

(a) 65

(b) 60

(c) 55

(d) 70

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) 65

(b) 60

(c) 55

(d) 70

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) 65

(b) 60

(c) 55

(d) 70

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) 65

(b) 60

(c) 55

(d) 70

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) 65

(b) 60

(c) 55

(d) 70

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) 65

(b) 60

(c) 55

(d) 70

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

Out of 2530 subscribers of a local newspaper, 1946 take thepaper daily and 820 take both the daily and Sunday editions.How many subscribe to the Sunday edition?

(a) 584

(b) 1404

(c) 820

(d) 1126

(e) 1710

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) 584

(b) 1404

(c) 820

(d) 1126

(e) 1710

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) 584

(b) 1404

(c) 820

(d) 1126

(e) 1710

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) 584

(b) 1404

(c) 820

(d) 1126

(e) 1710

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) 584

(b) 1404

(c) 820

(d) 1126

(e) 1710

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) 584

(b) 1404

(c) 820

(d) 1126

(e) 1710

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

A farmer has 100 eggs, but 3 are rotten. He selects a dozen atrandom for a customer.

In how many ways can this be done without choosing a rottenegg?

(b) 97

(c) C (100, 12)

(d) C (97, 12)

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(b) 97

(c) C (100, 12)

(d) C (97, 12)

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(b) 97

(c) C (100, 12)

(d) C (97, 12)

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(b) 97

(c) C (100, 12)

(d) C (97, 12)

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(b) 97

(c) C (100, 12)

(d) C (97, 12)

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(b) 97

(c) C (100, 12)

(d) C (97, 12)

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(b) 97

(c) C (100, 12)

(d) C (97, 12)

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

In how many ways can the customer get at most 1 rotten egg?

(a) C (97, 12) + 3 · C (97, 11)

(b) 100

(c) 3 · C (100, 11)

(d) 3 · C (97, 11)

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) C (97, 12) + 3 · C (97, 11)

(b) 100

(c) 3 · C (100, 11)

(d) 3 · C (97, 11)

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) C (97, 12) + 3 · C (97, 11)

(b) 100

(c) 3 · C (100, 11)

(d) 3 · C (97, 11)

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) C (97, 12) + 3 · C (97, 11)

(b) 100

(c) 3 · C (100, 11)

(d) 3 · C (97, 11)

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) C (97, 12) + 3 · C (97, 11)

(b) 100

(c) 3 · C (100, 11)

(d) 3 · C (97, 11)

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) C (97, 12) + 3 · C (97, 11)

(b) 100

(c) 3 · C (100, 11)

(d) 3 · C (97, 11)

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) C (97, 12) + 3 · C (97, 11)

(b) 100

(c) 3 · C (100, 11)

(d) 3 · C (97, 11)

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

The starting point is the desire to analyze a repetitive processthat is so intricate that a full-fledged model would be beyondany hope to quantitatively describe.

A good example to keep in mind is the toss of a standardsix-sided die.

The dynamical action by the contact forces at point of releaseis by itself so complicated that an adequate model of the tossis not attainable.

The compromise is to relinquish the desire to know thebehavior of any one instance of the process.

Instead the goal is to understand qualitatively how the processbehaves ‘in average’ when it is repeated many times.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

For the purposes here, assume the process results in only afinite number of outcomes.

The outcomes are collected into a set known as the samplespace.

In the toss-a-die process the sample space may be given asS = {1, 2, 3, 4, 5, 6}.

It is possible to use different labels for the outcomes, but themore obvious the connection is to the process the better.

When the process is subjected to the techniques about to bedeveloped it is referred to as a random experiment.

This phrase is somewhat misleading since the process need notbe thought of as an experiment, nor its evolution as random.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

Assume that n(S) = N where N is some integer greater than 1.

Denote the outcomes by S = {s1, . . . , sN}.

A probability distribution on S is a collection of numbers{p1, . . . , pN} where each number is in [0, 1] andp1 + · · ·+ pN = 1.

An event E is a subset of S , so E ⊂ S holds.

Write Pr(E ) for the probability of an event.

The definition is that Pr(E ) is the sum of all the pi with si ∈ E .

It follows that 0 ≤ Pr(E ) ≤ 1, Pr(∅) = 0, and Pr(S) = 1.

The following is a very useful fact: Pr(E c) = 1− Pr(E ).

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

If E and F are events, thenPr(E ∪ F ) = Pr(E ) + Pr(F )− Pr(E ∩ F ).

In particular, if E and F are mutually exclusive, thenPr(E ∪ F ) = Pr(E ) + Pr(F ).

If p1 = · · · = pN = 1N , then the probability distribution is said

to be uniform.

Calculations of probabilities are considerably simplified inspaces with uniform probability distribution.

In a sample space with uniform probability distribution

Pr(E ) =n(E )

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

to be uniform.

Pr(E ) =n(E )

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

to be uniform.

Pr(E ) =n(E )

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

to be uniform.

Pr(E ) =n(E )

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

to be uniform.

Pr(E ) =n(E )

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

A fair die has uniform probability distribution.

If E = ‘Toss 5 or 6’, then n(E ) = 2 and n(S) = 6 soPr(E ) = 2/6 = 1/3.

Suppose a die is ‘biased’, ‘unfair’, or ‘loaded’ so that p6 = 1/5.

Assume p1 = · · · = p5 = x so that 5x + 15 = 1.

It follows that 5x = 4/5 and x = 4/25.

This time Pr(E ) = 4/25 + 1/5 = 9/25, and 27/75 > 25/75,which is a subtle but noticeable difference.

In some circumstances it is possible to assign several differentsample spaces to the same random experiment.

The issue about to be presented is of concern in what is knownas statistical mechanics, a subject matter in physics.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

Consider the process of tossing two indistinguishable dicesimultaneously.

A reasonable sample space is given by

S = {{1, 1}, {1, 2}, . . . , {1, 6},

{2, 2}, {2, 3}, . . . , {2, 6},

{3, 3}, {3, 4}, {3, 5}, {3, 6},

{4, 4}, {4, 5}, {4, 6}, {5, 5}, {5, 6}, {6, 6}}.

Observe that n(S) = 6 + 5 + 4 + 3 + 2 + 1 = 21.

The interpretation of an outcome like {2, 5} is that one of thedice shows 2 and the other shows 5.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

S = {{1, 1}, {1, 2}, . . . , {1, 6},

{2, 2}, {2, 3}, . . . , {2, 6},

{3, 3}, {3, 4}, {3, 5}, {3, 6},

{4, 4}, {4, 5}, {4, 6}, {5, 5}, {5, 6}, {6, 6}}.

Observe that n(S) = 6 + 5 + 4 + 3 + 2 + 1 = 21.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

S = {{1, 1}, {1, 2}, . . . , {1, 6},

{2, 2}, {2, 3}, . . . , {2, 6},

{3, 3}, {3, 4}, {3, 5}, {3, 6},

{4, 4}, {4, 5}, {4, 6}, {5, 5}, {5, 6}, {6, 6}}.

Observe that n(S) = 6 + 5 + 4 + 3 + 2 + 1 = 21.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

S = {{1, 1}, {1, 2}, . . . , {1, 6},

{2, 2}, {2, 3}, . . . , {2, 6},

{3, 3}, {3, 4}, {3, 5}, {3, 6},

{4, 4}, {4, 5}, {4, 6}, {5, 5}, {5, 6}, {6, 6}}.

Observe that n(S) = 6 + 5 + 4 + 3 + 2 + 1 = 21.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

If it is assumed that both dice are fair, thenp1 = p7 = p12 = p16 = p19 = p21 since in each case the twodice show the same value.

It is also reasonable to assume that the probability value whenthe two dice show different values should also all be the same.

The intriguing question is whether p1 = p2.

In the case of dice it may be argued that a possibly very closeexamination of the two dice would reveal, however minuscule, adifference.

When dealing with gas molecules it becomes more of aphilosophical question whether such difference is present or not.

Such a difference, nonetheless, suggests that there are twodistinct ways to produce {1, 2}, whereas there is only a uniqueway to toss {1, 1}.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

Quantitatively this supports the claim that p2 = 2p1.

There are 6 probabilities with value p1 and 15 with value 2p1.

The total is 1 so p1 = 136 .

The event E = ‘Toss 11 or higher’ has probability236 + 1

36 = 336 = 1

When gas-molecules are considered as one does in statisticalmechanics, then it may well be questioned whether two distinctmolecules are distinguishable at any level of resolution.

As one ponders this kind of issue more deeply there is aconnection with the philosophical and religious concept ofidentity.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

36 = 336 = 1

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

36 = 336 = 1

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

36 = 336 = 1

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

36 = 336 = 1

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

36 = 336 = 1

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

Even when it is agreed that two objects, say a pair of hydrogenatoms, cannot be distinguish experimentally, there is still asense that each has its own identity.

In the present setting one takes a pragmatic view and thesample space S is given a probability distribution that is notuniform.

An alternative is to assume at the outset that the two dice aredistinguishable.

Assume one is red and the other is green.

This time the sample space looks like

S ′ = {(1, 1), . . . , (1, 6), (2, 1), . . . , (2, 6), . . . , (6, 1), . . . , (6, 6)}.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

S ′ = {(1, 1), . . . , (1, 6), (2, 1), . . . , (2, 6), . . . , (6, 1), . . . , (6, 6)}.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

S ′ = {(1, 1), . . . , (1, 6), (2, 1), . . . , (2, 6), . . . , (6, 1), . . . , (6, 6)}.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

S ′ = {(1, 1), . . . , (1, 6), (2, 1), . . . , (2, 6), . . . , (6, 1), . . . , (6, 6)}.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

S ′ = {(1, 1), . . . , (1, 6), (2, 1), . . . , (2, 6), . . . , (6, 1), . . . , (6, 6)}.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

An outcome like (2, 5) means specifically that the red die shows2 and the green die shows 5.

Observe that n(S ′) = 36.

The probability distribution is uniform, and each probabilityp1, . . . , p36 is 1

This time E = {(5, 6), (6, 5), (6, 6)}, and n(E ) = 3, soPr(E ) = 3

36 = 112 .

This value is equal to the value calculated when the probabilitydistribution is not uniform.

This vindicates the theory since the description of the event inthis case only involves the random experiment and not thesample space employed.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

36 = 112 .

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

36 = 112 .

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

36 = 112 .

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

36 = 112 .

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

36 = 112 .

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

This also illustrates that in some cases it possible to select asample space with uniform probability distribution although thisis not necessarily suggested by the random experiment itself.

If the option is available to employ a sample space withuniform probability distribution, then by all means take it sinceit will simplify the calculations.

Suppose some event is known to take place, then it is possibleto replace the original sample space with that event.

On the level of events one now only considers subsets of thatevent as the new events.

Events that were meaningful in the original sample space maystill be meaningful, but only if they intersect the given event.

The probabilities must be adjusted upwards to reflect that thenew sample space is inside the old sample space.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

If E is the event, then the adjustment is to multiply by 1Pr(E) .

For an event F the new probability is written Pr(F |E ) and it isgiven by

Pr(F |E ) =Pr(F ∩ E )

Pr(E ).

This is known as the conditional probability, but it may becalled the subset probability.

Observe that if E = S then the formula becomesPr(F |S) = Pr(F ).

The formula is not symmetric and Pr(E |F ) = Pr(F |E ) only ifPr(E ) = Pr(F ).

As expected, Pr(E |E ) = 1.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

Pr(F |E ) =Pr(F ∩ E )

Pr(E ).

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

Pr(F |E ) =Pr(F ∩ E )

Pr(E ).

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

Pr(F |E ) =Pr(F ∩ E )

Pr(E ).

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

Pr(F |E ) =Pr(F ∩ E )

Pr(E ).

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

Pr(F |E ) =Pr(F ∩ E )

Pr(E ).

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

The algebraic rewrite Pr(F ∩ E ) = Pr(E )Pr(F |E ) is usedfrequently.

Consider the unit square

S = {(x , y) ∈ R2 | 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1}.

Let S represent a sample space, and in the spirit of Venndiagrams let regions in the square represent events.

The area of S is 1, and let the area of a region correspond tothe probability of the corresponding event.

Let E be represented by the region left of the vertical linex = 1/2.

Let Fα be represented by the region below the line from (0, α)to (1, 1− α) where 0 ≤ α ≤ 1.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

S = {(x , y) ∈ R2 | 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1}.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

S = {(x , y) ∈ R2 | 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1}.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

S = {(x , y) ∈ R2 | 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1}.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

S = {(x , y) ∈ R2 | 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1}.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

S = {(x , y) ∈ R2 | 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1}.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

Observe that Pr(E ) = Pr(Fα) = 1/2.

The region Fα ∩ E may be split into a triangle and a rectangle.

Suppose α ≥ 1/2, then the rectangle has area 1/4 and thetriangle has area (α− 1

2)× 12 ×

It follows that Pr(Fα ∩ E ) = 2α+18 .

When α ≤ 1/2, then the rectangle has area α× 12 and the

triangle has area (12 − α)× 12 ×

It follows that Pr(Fα ∩ E ) = 2α+18 , which is the same formula

as before.

In either case it is true that Pr(Fα|E ) = 2α+14 .

When α = 0 one gets 14 , so the probability of Fα has been

reduced from 1/2 to 1/4.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

2)× 12 ×

as before.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

2)× 12 ×

as before.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

2)× 12 ×

as before.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

2)× 12 ×

as before.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

2)× 12 ×

as before.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

2)× 12 ×

as before.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

2)× 12 ×

as before.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

increased from 1/2 to 3/4.

In the case α = 1/2, then the probability of Fα has not beenchanged at all.

This last case is very exceptional but nonetheless ofconsiderable interest.

In general, an event F is said to be independent of and event Eif Pr(F |E ) = Pr(F ).

Suppose F is independent of E , then

Pr(E |F ) =Pr(E ∩ F )

Pr(F )=

Pr(E )Pr(F |E )

Pr(F )=

Pr(E )Pr(F )

Pr(F )= Pr(E ).

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

Pr(E |F ) =Pr(E ∩ F )

Pr(F )=

Pr(E )Pr(F |E )

Pr(F )=

Pr(E )Pr(F )

Pr(F )= Pr(E ).

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

Pr(E |F ) =Pr(E ∩ F )

Pr(F )=

Pr(E )Pr(F |E )

Pr(F )=

Pr(E )Pr(F )

Pr(F )= Pr(E ).

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

Pr(E |F ) =Pr(E ∩ F )

Pr(F )=

Pr(E )Pr(F |E )

Pr(F )=

Pr(E )Pr(F )

Pr(F )= Pr(E ).

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

Pr(E |F ) =Pr(E ∩ F )

Pr(F )=

Pr(E )Pr(F |E )

Pr(F )=

Pr(E )Pr(F )

Pr(F )= Pr(E ).

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

It follows that E is independent of F , so the concept ofindependence is symmetric.

Consider the experiment of drawing a ball from an urn with 1red ball and 2 blue balls and replacing it, and then drawing aball again.

Let E be the event ‘the first ball drawn is red’, and F the event‘the second ball drawn is red’.

The sample space is S = {(r , r), (r , b), (b, r), (b, b)}.

One may argue that p2 = 2p1, p3 = p2, and p4 = 2p2.

It must be that p1 + 2p1 + 2p1 + 4p1 = 1 so p1 = 1/9.

The event E has Pr(E ) = p1 + p2 = 3/9 = 1/3.

The event F has Pr(F ) = p1 + p3 = 1/3.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

The event F ∩ E has Pr(F ∩ E ) = 1/9.

It follows that

Pr(F |E ) =Pr(F ∩ E )

Pr(E )=

3= Pr(F ).

The events E and F are independent.

Observe that Pr(E ∩ F ) = Pr(E )Pr(F ) when the events areindependent.

It is immediate that the events ‘first ball drawn is blue’ and‘second ball drawn is blue’ are independent.

In fact any pair of the four events are independent.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

It follows that

Pr(F |E ) =Pr(F ∩ E )

Pr(E )=

3= Pr(F ).

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

It follows that

Pr(F |E ) =Pr(F ∩ E )

Pr(E )=

3= Pr(F ).

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

It follows that

Pr(F |E ) =Pr(F ∩ E )

Pr(E )=

3= Pr(F ).

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

It follows that

Pr(F |E ) =Pr(F ∩ E )

Pr(E )=

3= Pr(F ).

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

It follows that

Pr(F |E ) =Pr(F ∩ E )

Pr(E )=

3= Pr(F ).

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

If this fact is known before one calculates, thenp1 = 1

3 ×13 = 1

9 , p2 = 13 ×

23 = 2

9 , p3 = 23 ×

13 = 2

9 , and

This calculation hints at the utility of knowing independencebefore calculating.

There is an intuitive understanding of the concept ofindependence that may be appealed to first.

In the example one may argue that there is no way theoutcome of the first draw influences the probabilities of theoutcomes of the second draw.

The trouble is that the simplifying impact on the calculationstempts one to assume that the events involved are independent.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

3 ×13 = 1

9 , p2 = 13 ×

23 = 2

9 , p3 = 23 ×

13 = 2

9 , and

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

3 ×13 = 1

9 , p2 = 13 ×

23 = 2

9 , p3 = 23 ×

13 = 2

9 , and

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

3 ×13 = 1

9 , p2 = 13 ×

23 = 2

9 , p3 = 23 ×

13 = 2

9 , and

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

3 ×13 = 1

9 , p2 = 13 ×

23 = 2

9 , p3 = 23 ×

13 = 2

9 , and

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

For instance, are the events ‘mechanical failure of left wing jetengine’ and ‘mechanical failure of right wing jet engine’independent?

There may be a considerable amount of engineering designdirected towards a desire to make the two events independent.

This by itself does not guarantee that this is indeed the case.

If the example is changed so that there is no replacement afterthe first draw, then the sample space isS = {(r , b), (b, r), (b, b)}.

This time p1 = p2 = p3 = 1/3, Pr(E ) = Pr(F ) = 2/3, andPr(E ∩ F ) = 1/3.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

Pr(F |E ) =1323

26= Pr(F ).

This technical dependence also agrees with the intuitive ideathat the events here are not independent.

Do not confuse the notion of ‘mutual exclusive events’, whichmeans Pr(E ∪ F ) = Pr(E ) + Pr(F ), with ‘independent events’,which means Pr(E ∩ F ) = Pr(E )Pr(F ).

If E and F are independent, it is natural to ask if E c and F c

are independent, or if E and F c are independent.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

Pr(F |E ) =1323

26= Pr(F ).

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

Pr(F |E ) =1323

26= Pr(F ).

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

Pr(F |E ) =1323

26= Pr(F ).

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

First consider

Pr(F c |E c) =Pr(F c ∩ E c)

Pr(E c)=

Pr((F ∪ E )c)

Pr(E c)=

1− Pr(E ∪ F )

Pr(E c)=

1− Pr(E )− Pr(F ) + Pr(E ∩ F )

Pr(E c),

Pr(F c |E c) =1− Pr(E )− Pr(F ) + Pr(E )Pr(F )

1− Pr(E )=

1− Pr(F ) = Pr(F c).

Next consider

Pr(F c |E ) =Pr(F c ∩ E )

Pr(E )=

Pr(E )− Pr(F ∩ E )

Pr(E )=

Pr(E )− Pr(F )Pr(E )

Pr(E )= 1− Pr(F ) = Pr(F c).

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

First consider

Pr(F c |E c) =Pr(F c ∩ E c)

Pr(E c)=

Pr((F ∪ E )c)

Pr(E c)=

1− Pr(E ∪ F )

Pr(E c)=

1− Pr(E )− Pr(F ) + Pr(E ∩ F )

Pr(E c),

Pr(F c |E c) =1− Pr(E )− Pr(F ) + Pr(E )Pr(F )

1− Pr(E )=

1− Pr(F ) = Pr(F c).

Next consider

Pr(F c |E ) =Pr(F c ∩ E )

Pr(E )=

Pr(E )− Pr(F ∩ E )

Pr(E )=

Pr(E )− Pr(F )Pr(E )

Pr(E )= 1− Pr(F ) = Pr(F c).

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

It is definitely worthwhile to scrutinize these two derivations tomake sure that each step is understood.

They involve a nice mix of deMorgan’s law, and other basiclaws of set and probability theory.

To organize calculations that involve conditional probabilities itis useful to draw a tree diagram.

For simplicity, consider the application where D is a givenevent, with P and N subsequent events.

The tree diagram has an initial node from where two edgesemanate ending at a node marked D and a node marked Dc .

Both node D and node Dc have a pair of edges emanating totheir own pair of nodes marked + and −, respectively.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

To complete the tree we will add probabilities p and 1− p tothe first pair of edges, in order.

Next add 1− q and q to the pair emanating from D, as well asr and 1− r from Dc , each, in order.

Think of D as ‘having a particular disease’, and p theprobability of having the disease.

Think of + as ‘testing positive’ and − as ‘testing negative’.

The q is known as the probability of a false negative.

The r is known as the probability of a false positive.

In realistic applications all three numbers p, q, r are small.

It is also reasonable to as reasonable to assume that tests aredesigned so that q is smaller than r .

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

The following holds: Pr(D) = p, Pr(Dc) = 1− p,Pr(+|D) = 1− q,Pr(−|D) = q, Pr(+|Dc) = r , andPr(−|Dc) = 1− r .

There are therefore 6 probabilities indicated in the tree diagram.

The following 4 quantities are quickly calculated from data inthe tree diagram: Pr(D ∩+) = p(1− q), Pr(D ∩ −) = pq,Pr(Dc ∩+) = (1− p)r , and Pr(Dc ∩ −) = (1− p)(1− r).

With a little bit more work one calculatesPr(+) = Pr(D ∩+) + Pr(Dc ∩+) = p(1− q) + (1− p)r , andPr(−) = Pr(D ∩ −) + Pr(Dc ∩ −) = pq + (1− p)(1− r).

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

Finally, there are 4 more additional quantities one maycalculate:

Pr(D|+) =Pr(D ∩+)

Pr(+)=

p(1− q)

p(1− q) + (1− p)r,

Pr(D|−) =Pr(D ∩ −)

Pr(−)=

pq + (1− p)(1− r),

Pr(Dc |+) =Pr(Dc ∩+)

Pr(+)=

(1− p)r

p(1− q) + (1− p)r,

Pr(Dc |−) =Pr(Dc ∩ −)

Pr(−)=

(1− p)(1− r)

pq + (1− p)(1− r).

In summary, there are 16 probabilities one may extract from thetree diagram with only some additional basic calculations.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

Finally, there are 4 more additional quantities one maycalculate:

Pr(D|+) =Pr(D ∩+)

Pr(+)=

p(1− q)

p(1− q) + (1− p)r,

Pr(D|−) =Pr(D ∩ −)

Pr(−)=

pq + (1− p)(1− r),

Pr(Dc |+) =Pr(Dc ∩+)

Pr(+)=

(1− p)r

p(1− q) + (1− p)r,

Pr(Dc |−) =Pr(Dc ∩ −)

Pr(−)=

(1− p)(1− r)

pq + (1− p)(1− r).

In summary, there are 16 probabilities one may extract from thetree diagram with only some additional basic calculations.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

Suppose p = 2%, q = 1%, and r = 5%.

The exact formula is

Pr(D|+) =2(100− 1)

2(100− 1) + (100− 2)5=

198 + 490=

By ignoring products one gets the approximate formula

Pr(D|+) ≈ p

p + r=

2 + 5=

The other approximate formulas are

Pr(D|−) ≈ 0,

Pr(Dc |+) ≈ r

p + r,

Pr(Dc |−) ≈ 1.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

Suppose p = 2%, q = 1%, and r = 5%.

Pr(D|+) =2(100− 1)

2(100− 1) + (100− 2)5=

198 + 490=

Pr(D|+) ≈ p

p + r=

2 + 5=

Pr(D|−) ≈ 0,

Pr(Dc |+) ≈ r

p + r,

Pr(Dc |−) ≈ 1.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

Suppose p = 2%, q = 1%, and r = 5%.

Pr(D|+) =2(100− 1)

2(100− 1) + (100− 2)5=

198 + 490=

Pr(D|+) ≈ p

p + r=

2 + 5=

Pr(D|−) ≈ 0,

Pr(Dc |+) ≈ r

p + r,

Pr(Dc |−) ≈ 1.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

Suppose p = 2%, q = 1%, and r = 5%.

Pr(D|+) =2(100− 1)

2(100− 1) + (100− 2)5=

198 + 490=

Pr(D|+) ≈ p

p + r=

2 + 5=

Pr(D|−) ≈ 0,

Pr(Dc |+) ≈ r

p + r,

Pr(Dc |−) ≈ 1.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

The interest in these particular formulas is largely due to thefact that there is very little intuition regarding the values pfP(D|+) and P(Dc |+).

Rewrite the approximate formulas one more time in the forms

Pr(D|+) ≈ 1

1 + rp

Pr(Dc |+) ≈ 1pr + 1

Observe that Pr(D|+) + Pr(Dc |+) = 1 both in the exact andthe approximate case.

The approximate formulas supports the view that the crucialquantity in medical testing is the ratio of false positives to thelikelihood of having the disease.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

Pr(D|+) ≈ 1

1 + rp

Pr(Dc |+) ≈ 1pr + 1

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

Pr(D|+) ≈ 1

1 + rp

Pr(Dc |+) ≈ 1pr + 1

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

Pr(D|+) ≈ 1

1 + rp

Pr(Dc |+) ≈ 1pr + 1

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

This produces the odd circumstance where the societal concernthat sick people spread the disease when given false negativesfocuses the attention on q, whereas the individual’s desire toassess the likelihood of having the disease focuses the attentionon false positives and r .

If the false positive probability is equal to the probability ofhaving the disease, then both approximate values are 50%.

The exact formulas produce 1−q2−p−q and 1−p

2−p−q , so it is seenthat this way of ‘approximating’ is not necessarily completelymeaningful.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

The general tree diagram assumes the sample space has thepartition S = E1 ∪ · · · ∪ EK , where each pair Ei ,Ei ′ with i 6= i ′

consists of mutually exclusive events.

It follows that there are K branches with a probabilitydistribution p1, . . . , pK so that p1 + · · ·+ pK = 1.

At the second stage there are mutually exclusive eventsF i1, . . . ,F

iL and Li branches emanating from each Ei .

The reasoning illustrated in the case K = L = 2 generalizes,and the resulting formulas Pr(Ei |F i

j ), sometimes referred to asthe a priori probabilities, are given by the so-called BayesTheorem.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

When written in general form these formulas look intimidating,but if in specific case the procedure illustrated by theK = L = 2 case is employed, then these formulas follownaturally.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

What is the probability the customer gets at most 1 rotten egg?

(b)C (97, 12)

C (100, 12)

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(b)C (97, 12)

C (100, 12)

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(b)C (97, 12)

C (100, 12)

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(b)C (97, 12)

C (100, 12)

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(b)C (97, 12)

C (100, 12)

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(b)C (97, 12)

C (100, 12)

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(b)C (97, 12)

C (100, 12)

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

If E and F are events with Pr(E ) = .7, Pr(F ) = .4 andPr(E

⋂F ) = .3, what is Pr(E c

⋂F c)?

(a) .4

(b) .2

(c) .28

(d) .1

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

⋂F c)?

(a) .4

(b) .2

(c) .28

(d) .1

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

⋂F c)?

(a) .4

(b) .2

(c) .28

(d) .1

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

⋂F c)?

(a) .4

(b) .2

(c) .28

(d) .1

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

⋂F c)?

(a) .4

(b) .2

(c) .28

(d) .1

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

⋂F c)?

(a) .4

(b) .2

(c) .28

(d) .1

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

An exam has eight True/False questions. What is theprobability that a student guessing will get at least one problemcorrect?

(d)255

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(d)255

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(d)255

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(d)255

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(d)255

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(d)255

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

I develop a test for a disease that affects .01% of thepopulation. If a person has the disease my test will be positive80% of the time. If a person does not have the disease my testwill be negative 70% of the time.

Approximately, what is the probability that a randomly chosenperson tests positive?

(a) .01

(b) .02

(c) .3

(d) .05

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) .01

(b) .02

(c) .3

(d) .05

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) .01

(b) .02

(c) .3

(d) .05

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) .01

(b) .02

(c) .3

(d) .05

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) .01

(b) .02

(c) .3

(d) .05

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) .01

(b) .02

(c) .3

(d) .05

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) .01

(b) .02

(c) .3

(d) .05

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

If you test positive, approximately what is the probability thatyou have the disease?

(a) .3

(b) .0005

(c) .01

(d) .0003

(e) .7

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) .3

(b) .0005

(c) .01

(d) .0003

(e) .7

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) .3

(b) .0005

(c) .01

(d) .0003

(e) .7

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) .3

(b) .0005

(c) .01

(d) .0003

(e) .7

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) .3

(b) .0005

(c) .01

(d) .0003

(e) .7

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) .3

(b) .0005

(c) .01

(d) .0003

(e) .7

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

A pair of fair dice is cast. What is the probability that at leastone four is cast?

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

Let A and B be events in a sample space S such that:

Pr(A) = .6 Pr(B) = .5 Pr(A⋂

B) = .2

Find Pr(A⋃

(a) 1.1

(b) .9

(c) .6

(d) .4

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

Pr(A) = .6 Pr(B) = .5 Pr(A⋂

B) = .2

Find Pr(A⋃

(a) 1.1

(b) .9

(c) .6

(d) .4

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

Pr(A) = .6 Pr(B) = .5 Pr(A⋂

B) = .2

Find Pr(A⋃

(a) 1.1

(b) .9

(c) .6

(d) .4

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

Pr(A) = .6 Pr(B) = .5 Pr(A⋂

B) = .2

Find Pr(A⋃

(a) 1.1

(b) .9

(c) .6

(d) .4

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

Pr(A) = .6 Pr(B) = .5 Pr(A⋂

B) = .2

Find Pr(A⋃

(a) 1.1

(b) .9

(c) .6

(d) .4

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

Pr(A) = .6 Pr(B) = .5 Pr(A⋂

B) = .2

Find Pr(A⋃

(a) 1.1

(b) .9

(c) .6

(d) .4

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

Pr(A) = .6 Pr(B) = .5 Pr(A⋂

B) = .2

Find Pr(A⋃

(a) 1.1

(b) .9

(c) .6

(d) .4

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

A card is picked from a deck. Let E be the event ‘a club ispicked’ and F the event ‘an ace is picked’. Which of thefollowing is false?

(a) Pr(E ) = 14

(b) E and F are independent

(c) E and F are mutually exclusive

(d) Pr(F ) = 113

(e) None of these.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) Pr(E ) = 14

(d) Pr(F ) = 113

(e) None of these.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) Pr(E ) = 14

(d) Pr(F ) = 113

(e) None of these.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) Pr(E ) = 14

(d) Pr(F ) = 113

(e) None of these.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) Pr(E ) = 14

(d) Pr(F ) = 113

(e) None of these.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) Pr(E ) = 14

(d) Pr(F ) = 113

(e) None of these.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

A biased coin is flipped 3 times. If the coin comes up heads60% of the time, what is the probability of getting 3 tails?

(a) .216

(b) .192

(c) .064

(d) .125

(e) .4

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) .216

(b) .192

(c) .064

(d) .125

(e) .4

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) .216

(b) .192

(c) .064

(d) .125

(e) .4

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) .216

(b) .192

(c) .064

(d) .125

(e) .4

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) .216

(b) .192

(c) .064

(d) .125

(e) .4

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) .216

(b) .192

(c) .064

(d) .125

(e) .4

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

If a fair coin is flipped 7 times, what is the probability ofgetting 4 heads and 3 tails?

(a) 35128

(b) 1128

(c) 52128

(d) 210128

(e) 116

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) 35128

(b) 1128

(c) 52128

(d) 210128

(e) 116

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) 35128

(b) 1128

(c) 52128

(d) 210128

(e) 116

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) 35128

(b) 1128

(c) 52128

(d) 210128

(e) 116

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) 35128

(b) 1128

(c) 52128

(d) 210128

(e) 116

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) 35128

(b) 1128

(c) 52128

(d) 210128

(e) 116

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

At a particular university, it is found that 60% of students thatpass college algebra go on to pass calculus I. However, only10% of those that do not pass college algebra go on to passcalculus I. If 55% pass college algebra, what percentage ofstudents pass calculus I?

(a) 33%

(b) 37.5%

(c) 54%

(d) 23.5%

(e) 32.5%

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) 33%

(b) 37.5%

(c) 54%

(d) 23.5%

(e) 32.5%

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) 33%

(b) 37.5%

(c) 54%

(d) 23.5%

(e) 32.5%

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) 33%

(b) 37.5%

(c) 54%

(d) 23.5%

(e) 32.5%

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) 33%

(b) 37.5%

(c) 54%

(d) 23.5%

(e) 32.5%

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) 33%

(b) 37.5%

(c) 54%

(d) 23.5%

(e) 32.5%

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

Suppose E and F are mutually exclusive events with P(E ) = .2and P(F ) = .5. What is P(E c ∩ F c)?

(a) .15

(c) .3

(d) .9

(e) .7

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) .15

(c) .3

(d) .9

(e) .7

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) .15

(c) .3

(d) .9

(e) .7

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) .15

(c) .3

(d) .9

(e) .7

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) .15

(c) .3

(d) .9

(e) .7

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) .15

(c) .3

(d) .9

(e) .7

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

If one card is drawn from a 52 card deck, what is theprobability it is an ace or a heart?

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

If a multiple choice exam is given with 15 questions and 5possible answers per question, what is the probability thatrandom guessing will answer all equations incorrectly?

(a) .0352

(b) 13

(e) .542

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) .0352

(b) 13

(e) .542

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) .0352

(b) 13

(e) .542

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) .0352

(b) 13

(e) .542

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) .0352

(b) 13

(e) .542

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) .0352

(b) 13

(e) .542

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

Given two events, E and F , with P(E ) = .4,P(F |E ) = .8 andP(F ) = .7, what is P(E c ∩ F c)?

(a) .22

(b) .9

(c) .46

(d) .5

(e) This situation is impossible.

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) .22

(b) .9

(c) .46

(d) .5

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) .22

(b) .9

(c) .46

(d) .5

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) .22

(b) .9

(c) .46

(d) .5

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) .22

(b) .9

(c) .46

(d) .5

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) .22

(b) .9

(c) .46

(d) .5

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

Given independent events, E and F , with P(E ) = .3 andP(F ) = .4, what is P(F ∩ E c)?

(a) .7

(b) .3

(c) .28

(d) .12

(e) .18

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) .7

(b) .3

(c) .28

(d) .12

(e) .18

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) .7

(b) .3

(c) .28

(d) .12

(e) .18

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) .7

(b) .3

(c) .28

(d) .12

(e) .18

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) .7

(b) .3

(c) .28

(d) .12

(e) .18

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) .7

(b) .3

(c) .28

(d) .12

(e) .18

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

Suppose .1% of a population has John-Jacobs disease and atest is 99.5% accurate (meaning it is correct 99.5% of thetime). If someone tests positive for John-Jacobs disease, whatis the probability they actually have it?

(a) .834

(b) .995

(c) .005

(d) .462

(e) .166

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) .834

(b) .995

(c) .005

(d) .462

(e) .166

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) .834

(b) .995

(c) .005

(d) .462

(e) .166

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) .834

(b) .995

(c) .005

(d) .462

(e) .166

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) .834

(b) .995

(c) .005

(d) .462

(e) .166

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) .834

(b) .995

(c) .005

(d) .462

(e) .166

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

I choose 3 cards from a standard deck of 52 cards. What is theprobability they are all red?

(a) .0577

(b) .118

(c) .216

(d) .125

(e) .083

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) .0577

(b) .118

(c) .216

(d) .125

(e) .083

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) .0577

(b) .118

(c) .216

(d) .125

(e) .083

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) .0577

(b) .118

(c) .216

(d) .125

(e) .083

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) .0577

(b) .118

(c) .216

(d) .125

(e) .083

FiniteMathematics

Linner

Set Theory

Finite sets.

Set notation.

Subsets.

Sets of sets.

Set-operations.

Examples.

DeMorgan’slaws.

Proof I.

Proof II.

Venn-diagram I.

Venn-diagramII.

Exclusive OR.

Counting.

Countingsubsets.

Notation

Counting

Countingsubsets.

(a) .0577

(b) .118

(c) .216

(d) .125

(e) .083

Finite Linn er Set Theory Finite MathematicsFinite Mathematics Linn er Set Theory Finite sets. Set...

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