1Melikyan/DM/Fall09

Discrete Mathematics

Ch. 5 Sets (Review)

Instructor: Hayk Melikyanmelikyan@nccu.edu

Today we will review sections 5.1, 5.2, 5.3

2Melikyan/DM/Fall09

What is a set?

A collection of elements:Order is irrelevantNo repetitionsCan be infiniteCan be empty

Examples:{Angela, Belinda, Jean}{0,1,2,3,…}

3Melikyan/DM/Fall09

Operations on sets

S is a set Membership:xS x is an element of S Angela{Angela, Belinda, Jean}

Subset S1 S – Set S1 is a subset of set S– All elements of S1 are elements of S– {Angela,Belinda} {Angela, Belinda, Jean}

Proper subset S1 S

4Melikyan/DM/Fall09

Operations on sets

If S, S1 are sets

Intersection: S S1

– is a set of all elements that belong to both

{Ang, Bel, Jea} {Ang, Dan} = {Ang}

Union: S S1

– is a set of all elements that belong to either

– {Ang, Bel, Jea} {Ang, Dan} = {Ang, Bel, Jea, Dan}

5Melikyan/DM/Fall09

Operations on sets

Let S, S1 be sets

Equality: S = S1

– iff they have the same elements

Difference: S \ S1

– is a set of all elements that belong to S but NOT to S1

{Ang, Bel, Jea} \ {Ang, Dan} = {Bel, Jea}

6Melikyan/DM/Fall09

More notation

In mathematics sets are often specified with a predicate and an enveloping set as follows:

S = {xA | P(x)}

S is the set of all elements of A that satisfy predicate P

Example:

Q={xR | a,bZ b0 & x=a/b}

7Melikyan/DM/Fall09

Set Equality

Two sets are equal iff they have the same elements

Theorem: for any sets A and B, A=B iff AB & BA

8Melikyan/DM/Fall09

Book example 5.1.5

– 2{1,2,3} ?– {2}{1,2,3} ?– 2{1,2,3} ?– {2}{1,2,3} ?– {2}{{1},{2}} ?– {2}{{1},{2}} ?

How about set A such that {2} is a subset of it and A is an element of it?– A={1,2,{1},{2}}

9Melikyan/DM/Fall09

Universal SetIf we are dealing with sets which are all

subsets ofa larger set U then we call it a universal set U

All of your sets will be subsets of U

When does such a U exist?

Always, for we can set U to the union of all sets involved ???????

10Melikyan/DM/Fall09

Complement

So if I am dealing with set A which is a subset of the universal set U then:

I can define complement of A: AC = U\AThat is the set of all elements (of U) that are not in AOften “of U” is dropped and people say that AC isthe set of everything that is not in A

What is the complement of U? UC = Ø

What set has U as its complement? ØC=U

11Melikyan/DM/Fall09

Sets & Predicate LogicAll of the set operations and relations above

can be defined in terms of Boolean connectives:– AB={x | xA v xB}– AB={x | xA & xB}– A\B={x | xA & not xB}– AC={x | not xA}

– A = B iff x( xA xB)– A B iff x (xA xB)– AB iff x (xA xB) & not A=B

12Melikyan/DM/Fall09

Symmetric Difference

Set C is the symmetric difference of sets A and

B iff every element of C belongs to A or B butnot both

ABC [C=A B a (aC (aA xor aB))]

If A={1,2}, B={2,3} then A B={1,3}

In general: A A = {}

13Melikyan/DM/Fall09

Exercise 2Intersection of two sets is contained in their

union: AB [ (A B) (A B) ]Proof:

14Melikyan/DM/Fall09

Exercise 3

Union is commutativeAB [ A B = B A ]

Intersection is commutative

AB [ A B = B A ]

Intersection distributes over union:

ABC [ A (B C) =(A B) (A C) ]

15Melikyan/DM/Fall09

Exercise

In Exercise #5 we proved: ABC [ A (B C) =(A B) (A C) ]

using the fact that A&(BvC) = (A&B)v(A&C)Given the statement just proved

Av(B&C) = (AvB) & (AvC)

what can we now prove in terms of sets?Union distributes over intersection:

ABC [ A (B C) =(A B) (A C) ]

16Melikyan/DM/Fall09

Proof :

17Melikyan/DM/Fall09

Logic - Sets

v & avb=bva ab=ba a&b=b&a ab=ba (avb)vc=av(bvc) (ab)c=a(bc) (a&b)&c=a&(b&c) (ab)c=a(bc) a&(bvc)=(a&b)v(a&c)

a(bc)=(ab)(ac) av(b&c)=(avb) & (avc)

a(bc)=(ab)(ac)

18Melikyan/DM/Fall09

Cartesian Products

Intuition first:Suppose I have a function that takes two

numbers x and y and returns x/yWhat is the set of valid inputs?Is it just R?

– No -- cannot divide by 0

Is it R\{0}?– No -- can happily have 0 as x

19Melikyan/DM/Fall09

CombinationsSuppose I have:• two independent attributes:

sky conditions and temperaturetwo values for the sky conditions

S={sunny, overcast};three values for the precipitation:

P={snow, rain, nothing}.

How many combinations can I have?

<sunny, rain> , <sunny, snow> . <sunny, nothing>.

<overcast, rain>. <overcast, snow>, <overcast, nothing>

20Melikyan/DM/Fall09

Cartesian Product

Set C is a Cartesian Product of set A and set B iff it is a set of all ordered pairs such that the 1st element belongs to A and the 2nd element belongs to B

C=A B iff ab (<a, b>C (aA & bB))]

21Melikyan/DM/Fall09

Examples

A={0,1}, B={Ang, Bel}AB = {<0,Ang>, <0,Bel>, <1,Ang>, <1,

Bel>}

A={0,1}, B={Ang, Bel}BA = {<Ang, 0>, <Bel, 0>, <Ang, 1>,

<Bel,1>}

A={0}, B={a,b}, C={1,2}ABC={<0, a, 1>,<0, b, 1>, <0, a, 2>,<0,

b, 2>}

22Melikyan/DM/Fall09

More examples

A=B=C=D=R (set of all real numbers)ABCD=R4 (time-space continuum)

What is the cardinality of Cartesian Product?

|A1 … An|=|A1| · … · |An| for finite sets

How about { } {1,2}?– {} {1,2}={}