Discrete Structure

Sets

2

Set Theory• Set: Collection of objects (“elements”)

• aA “a is an element of A” “a is a member of A”

• aA “a is not an element of A”

• A = {a1, a2, …, an} “A contains…”

• Order of elements is meaningless

• It does not matter how often the same element is listed.

SetsCurly braces “{“ and “}” are used to denote

sets.Java note: In Java curly braces denote

arrays, a data-structure with inherent ordering.

Mathematical sets are unordered so different from Java arrays. Java arrays require that all elements be of the same type.

Mathematical sets don’t require this, however. EG: { }

Standard Numerical SetsThe natural numbers:N = { 0, 1, 2, 3, 4, … }

The integers:Z = { … -3, -2, -1, 0, 1, 2, 3, … }

The positive integers:Z+ = {1, 2, 3, 4, 5, … }

The real numbers: R --contains any decimal number of arbitrary precision The rational numbers: Q --these are decimal numbers whose decimal expansion repeats

Q: Give examples of numbers in R but not Q.

Set Theory

(a) Z=the set of integers={0,1,-1,2,-1,3,-3,...}(b) N=the set of nonnegative integers or natural numbers(c) Z+=the set of positive integers(d) Q=the set of rational numbers={a/b| a,b is integer, b not zero}(e) Q+=the set of positive rational numbers(f) Q*=the set of nonzero rational numbers(g) R=the set of real numbers(h) R+=the set of positive real numbers(i) R*=the set of nonzero real numbers(j) C=the set of complex numbers

-NotationThe Greek letter “” (epsilon) is used to denote that

an object is an element of a set. When crossed out “” denotes that the object is not an element.”

EG: 3 S reads:“3 is an element of the set S ”.

Q: Which of the following are true:1. 3 R2. -3 N3. -3 R4. 0 Z+

5. x xR x2=-5

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Examples for Sets• A = or A={ } “empty set/null

set”

• A = {z} Note: zA, but z {z}

• A = {{b, c}, {c, x, d}}

• A = {{x, y}} Note: {x, y} A, but {x, y} {{x, y}}

• A = {x | P(x)}“set of all x such that P(x)”

• A = {x | xN x > 7} = {8, 9, 10, …}“set builder notation”

Set Builder NotationUp to now sets have been defined using the

curly brace notation “{ … }” or descriptively “the set of all natural numbers”. The set builder notation allows for concise definition of new sets. For example { x | x is an even integer } { 2x | x is an integer }

are equivalent ways of specifying the set of all even integers.

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Set Operations

Union: AB = {x | xA xB}

Example: A = {a, b}, B = {b, c, d} AB = {a, b, c, d}

Intersection: AB = {x | xA xB}

Example: A = {a, b}, B = {b, c, d} AB = {b}

Set Builder Notation.Examples.

A1: U = N. { x | y (y x ) } = { 0 }A2: U = Z. { x | y (y x ) } = { }

A3: U = Z. { x | y (y R y 2 = x )}

= { 0, 1, 2, 3, 4, … } = N

A4: U = Z. { x | y (y R y 3 = x )} = Z

A5: U = R. { |x | | x Z } = NA6: U = R. { |x | } = non-negative reals.

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-NotationDEF: A set S is said to be a subset of the set T iff

every element of S is also an element of T. This situation is denoted by

S TA synonym of “subset” is “contained by”.Definitions are often just a means of establishing

a logical equivalence which aids in notation. The definition above says that:

S T x (xS ) (xT )We already had all the necessary concepts, but

the “” notation saves work.

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-NotationWhen “” is used instead of “”, proper

containment is meant. A subset S of T is said to be a proper subset if S is not equal to T. Notationally:

S T S T x (x S xT )

Q: What algebraic symbol is reminiscent of?

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SubsetsUseful rules:

• A = B (A B) (B A) • (A B) (B C) A C (see Venn Diagram)

UU

AABB CC

Union

AB = { x | x A x B }

Elements in at least one of the two sets:

A B

U

AB

Intersection

AB = { x | x A x B }

Elements in exactly one of the two sets:

A B

U

AB

Disjoint SetsDEF: If A and B have no common elements, theyare said to be disjoint, i.e. A B = .

A B

U

Set Difference

AB = { x | x A x B }

Elements in first set but not second:

A

B

U

AB

Symmetric Difference

A B

UAB

AB = { x | x A x B }

Elements in exactly one of the two sets:

Complement

A = { x | x A }

Elements not in the set (unary operator):

A

U

A

Set Identities

Identity laws Domination laws Idempotent laws Double complementation Commutativity Associativity Distribuitivity DeMorgan

logical identities rewriting as follows: disjunction “” becomes union “” conjunction “” becomes intersection “” negation “” becomes complementation “–” false “F” becomes the empty set true “T” becomes the universe of reference U

Set Identities

In fact, the logical identities create the set identities by applying the definitions of the various set operations. For example:

LEMMA: (Associativity of Unions) (AB )C = A(B C )

Proof : (AB )C = {x | x A B x C } (by def.)= {x | (x A x B ) x C } (by def.)= {x | x A ( x B x C ) } (logical

assoc.)= {x | x A (x B C ) } (by def.)= A(B C ) (by def.)

�Other identities are derived similarly.

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Visual DeMorgan

=BA

BA

Cardinality

The cardinality of a set is the number of distinct elements in the set. |S | denotes the cardinality of S.

Q: Compute each cardinality.1. |{1, -13, 4, -13, 1}|2. |{3, {1,2,3,4}, }|3. |{}|4. |{ {}, {{}}, {{{}}} }|

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Cardinality

Hint: After eliminating the redundancies just look at the number of top level commas and add 1 (except for the empty set).

A:1. |{1, -13, 4, -13, 1}| = |{1, -13, 4}| = 32. |{3, {1,2,3,4}, }| = 3. To see this, set S =

{1,2,3,4}. Compute the cardinality of {3,S, }

3. |{}| = || = 04. |{ {}, {{}}, {{{}}} }|

= |{ , {}, {{}}| = 3

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Cardinality

DEF: The set S is said to be finite if its cardinality is a nonnegative integer. Otherwise, S is said to be infinite.

EG: N, Z, Z+, R, Q are each infinite.

Note: We’ll see later that not all infinities are the same. In fact, R will end up having a bigger infinity-type than N, but surprisingly, N has same infinity-type as Z, Z+, and Q.

Set Theory

Counting and Venn Diagrams

In a class of 50 college freshmen, 30 are studyingBASIC, 25 studying PASCAL, and 10 are studying both. How

many freshmen are studying either computer language?

U A B

10 1520

5| | | | | | | |A B A B A B

Set Theory

Counting and Venn Diagrams

Given 100 samplesset A: with D1

set B: with D2

set C: with D3

Defect types of an AND gate:D1: first input stuck at 0

D2: second input stuck at 0D3: output stuck at 1

with |A|=23, |B|=26, |C|=30,

| | , | | , | | ,| |A B A C B CA B C

7 8 103 , how many samples have defects?

A

B

C

11 4

35

7

12

15

43

Ans:57

| | | | | | | | | || | | | | |A B C A B C A B

A C B C A B C

Power Set

DEF: The power set of S is the set of all subsets of S.

Denote the power set by P (S ) or by 2s .

The latter weird notation comes from the following.

Lemma: | 2s | = 2|s|

Power Set –ExampleTo understand the previous fact consider

S = {1,2,3}Enumerate all the subsets of S :0-element sets: {} 1 1-element sets: {1}, {2}, {3} +32-element sets: {1,2}, {1,3}, {2,3} +33-element sets: {1,2,3} +1Therefore: | 2s | = 8 = 23 = 2|s|

Cartesian ProductThe most famous example of 2-tuples are points

in the Cartesian plane R2. Here ordered pairs (x,y) of elements of R describe the coordinates of each point. We can think of the first coordinate as the value on the x-axis and the second coordinate as the value on the y-axis.

DEF: The Cartesian product of two sets A and B –denoted by A B– is the set of all ordered pairs (a, b) where aA and bB .

Q: Describe R2 as the Cartesian product of two sets.

Cartesian ProductA: A = {1,2}, B = {3,4}, C = {5,6,7}A B C ={ (1,3,5), (1,3,6), (1,3,7),

(1,4,5), (1,4,6), (1,4,7), (2,3,5), (2,3,6), (2,3,7), (2,4,5), (2,4,6), (2,4,7) }

Lemma: The cardinality of the Cartesian product is the product of the cardinalities:

| A1 A2 … An | = |A1||A2| … |An| Q: What does S equal?

Set Theory

Set Operations and the Laws of Set Theory

The Laws of Set Theory

A)BA(A

Laws Absorption A)BA(A (10)

Laws Domination =A ,UUA (9)

Laws Inverse AA ,UAA (8)

Laws Identity AUA ,AA (7)

Laws Idempotent AAA ,AAA (6)

Set Theory

Set Operations and the Laws of Set Theory

The Laws of Set Theory

)()()(

Laws )()()( (5)

)()(

Laws )()( (4)

Laws (3)

Laws ' (2)

of Law )1(

CABACBA

veDistributiCABACBA

CBACBA

eAssociativCBACBA

ABBA

eCommutativABBA

BABA

sDemorganBABA

ComplementDoubleAA

Generalized Union

Binary union operator: ABn-ary union:AA2…An : ((…((A1 A2) …) An)(grouping & order is irrelevant)“Big U” notation:

Or for infinite sets of sets:

n

iiA

1

XA

A

Generalized Intersection

Binary intersection operator: ABn-ary intersection:AA2…An((…((A1A2)…)An)(grouping & order is irrelevant)“Big Arch” notation:

Or for infinite sets of sets:

n

iiA

1

XA

A

Mutual subsetsExample: Show A(BC)=(AB)(AC).

Show A(BC)(AB)(AC). Assume xA(BC), & show x(AB)(AC). We know that xA, and either xB or xC.

Case 1: xB. Then xAB, so x(AB)(AC). Case 2: xC. Then xAC , so x(AB)(AC).

Therefore, x(AB)(AC). Therefore, A(BC)(AB)(AC).

Show (AB)(AC) A(BC). …

Membership Table Exercise

Prove (AB)C = (AC)(BC).A B C AABB ((AABB)) CC AA CC BB CC ((AA CC))((BB CC))0 0 00 0 10 1 00 1 11 0 01 0 11 1 01 1 1

Ordered n-tuplesNotationally, n-tuples look like sets except

that curly braces are replaced by parentheses: ( 11, 12 ) –a 2-tuple aka ordered pair ( , , ) –a 3-tuple ( , , , 11, Leo ) –a 5-tuple

Java: n -tuples are similar to Java arrays “{…}”, except that type-mixing isn’t allowed in Java.As opposed to sets, repetition and ordering do matter with n-tuples. (11, 11, 11, 12, 13) ( 11, 12, 13 ) ( , , ) ( , , )

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Exercises

Question 1:

Given a set A = {x, y, z} and a set B = {1, 2, 3, 4}, what is the value of | 2A 2B | ?

Question 2:

Is it true for all sets A and B that (AB)(BA) = ?Or do A and B have to meet certain conditions?

Question 3:

For any two sets A and B, if A – B = and B – A = , can we conclude that A = B? Why or why not?

Blackboard Exercise

Prove the following:If A B and B C then A C .

Set Theory

BABABA

ABxAxxBA

BA

B }|{

. Negate Ex.

BABA

BABABAAB

BBAABABABA

BABABABABA

BABABABA

BAxBAxxBA

BA

)()()()(

])[(])[()()(

)()()(

)()()()(

}|{

. Negate Ex.