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Set Theory. Dr. Ahmed Elmoasry. Contents. Ch I: Experiments, Models, and Probabilities. Ch II: Discrete Random Variables Ch III: Discrete Random Variables. Ch I: Experiments, Models, and Probabilities. Set Theory Applying set theory to probability Probability Axioms - PowerPoint PPT Presentation
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Set Theory
Dr. Ahmed Elmoasry
Contents
Ch I: Experiments, Models, and Probabilities.
Ch II: Discrete Random Variables
Ch III: Discrete Random Variables
Ch I: Experiments, Models, and Probabilities.
1. Set Theory2. Applying set theory to probability3. Probability Axioms4. Some consequences of the Axioms5. Conditional probability6. Independence7. sequential Experiments and tree diagrams8. Counting Methods9. Independent trials
Sets
Definition. A Set is any well defined collection of “objects.”Definition. The elements of a set are the objects in a set.Notation. Usually we denote sets with upper-case letters, elements with lower-case letters. The following notation is used to show set membership
means that x is a member of the set A means that x is not a member of the set
A.
AxAx
Ways of Describing Sets
List the elements
Give a verbal description
“A is the set of all integers from 1 to 6, inclusive”
Give a mathematical inclusion rule
A= 1,2,3,4,5,6
A= Integers 1 6x x
Examples of Sets
A={MUJ university, QAS University, the planet Mercury}
B={x2 | x =1,2,3,… }
C={all students who weigh more than 60 kg}
I={all positive integers, negative integers, and 0}
Set Equality
We say that A=B iff B A and A B
Or
A={0,17,46} B= {46,0,17} C={17,0,46}
BxAx
Some Special Sets
The Null Set or Empty Set. This is a set with no elements, often symbolized by
The Universal Set. This is the set of all elements currently under consideration, and is often symbolized by
Membership Relationships
Definition. Subset.
“A is a subset of B” We say “A is a subset of B” if , i.e.,
all the members of A are also members of B. The notation for subset is very similar to the notation for “less than or equal to,” and means, in terms of the sets, “included in or equal to.”
A B
x A x B
S x6 B x1 x2 x3 x5 A x4
x7
Membership Relationships
Definition. Proper Subset.
“A is a proper subset of B” We say “A is a proper subset of B” if all the
members of A are also members of B, but in addition there exists at least one element c such that but . The notation for subset is very similar to the notation for “less than,” and means, in terms of the sets, “included in but not equal to.”
A B
c B c A
Combining Sets – Set Union
“A union B” is the set of all elements that are in A, or B, or both.
This is similar to the logical “or” operator.
A B B A
Combining Sets – Set Intersection
“A intersect B” is the set of all elements that are in both A and B.
This is similar to the logical “and”
A B
Ex. Of set union
A is the set of student, who weigh more than 65 kg.
B is the set of student, who tall more than 165 cm.
A x3 B x1 x2 x4 x5 x6
x7 x8 x9 x10
Set Complement
“A complement,” or “not A” is the set of all elements not in A.
The complement operator is similar to the logical not, and is reflexive, that is,
A
A A
A
A
Set Difference
The set difference “A minus B” is the set of elements that are in A, with those that are in B subtracted out. Another way of putting it is, it is the set of elements that are in A, and not in B, so
A B
A B A B
A B
Examples
{1,2,3}A {3,4,5,6}B
{3}A B {1,2,3,4,5,6}A B
{1,2,3,4,5,6}
{4,5,6}B A {1,2}B
Venn Diagrams
Venn Diagrams use topological areas to stand for sets. I’ve done this one for you.
A B
A B
Venn Diagrams
Try this one!
A B
A B
Venn Diagrams
Here is another one
A B
A B
Mutually Exclusive and Exhaustive Sets
Definition. We say that a group of sets is exhaustive of another set if their union is equal to that set. For example, if we say that A and B are exhaustive with respect to C.
Definition. We say that two sets A and B are mutually exclusive if , that is, the sets have no elements in common.
A B C
A B
Mutually Exclusive Sets
A and B are disjoint iff A∩ B =
A
B
Set Partition
Definition. We say that a group of sets partitions another set if they are mutually exclusive and exhaustive with respect to that set. When we “partition a set,” we break it down into mutually exclusive and exhaustive regions, i.e., regions with no overlap. The Venn diagram below should help you get the picture. In this diagram, the set A (the rectangle) is partitioned into sets W,X, and Y.
Set Partition
A
WX Y
De Morgan’s Law
BABA )(
BABA )(
Some Test Questions
A ?
Some Test Questions
A A=?
Some Test Questions
A ?
Some Test Questions
A A=?
Some Test Questions
A A ?
Some Test Questions
A ?
Some Test Questions
?
Some Test Questions
If A B then A B ?
Some Test Questions
If A B then A B ?
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