August 2006 1

August 2006 2

Definitions

• A set is a collection of objects.

• Objects in the collection are called elements of the set.

August 2006 3

Examples - set The collection of persons living in INDIA is

a set.– Each person living in INDIA is an element of

the set.

The collection of all countries in the state of Texas is a set.– Each county in Texas is an element of the set.

August 2006 4

Examples - set The collection of counting numbers is a

set.– Each counting number is an element of the

set.

The collection of pencils in your briefcase is a set.– Each pencil in your briefcase is an element

of the set.

August 2006 5

Notation • Sets are usually designated with

capital letters.

• Elements of a set are usually designated with lower case letters.

August 2006 6

Notation• The roster method of specifying a

set consists of surrounding the collection of elements with braces.

For example the set of counting numbers from 1 to 5 would be written as

{1, 2, 3, 4, 5}.

August 2006 7

Notation• Set builder notation has the general

form

{variable | descriptive statement }. The vertical bar (in set builder notation) is always

read as “such that”.

Set builder notation is frequently used when the roster method is either inappropriate or inadequate.

August 2006 8

Example – set builder notation

{x | x < 6 and x is a counting number} is the set of all counting numbers less than 6. Note this is the same set as {1,2,3,4,5}.

{x | x is a fraction whose numerator is 1 and whose denominator is a counting number }.

August 2006 9

Definition• The set with no elements is called

the empty set or the null set and is designated with the symbol .

August 2006 10

Examples – empty set– The set of all pencils in your briefcase might indeed be the empty set.

– The set of even prime numbers greater than 2 is the empty set.

– The set {x | x < 3 and x > 5} is the empty set.

August 2006 11

Definition - subset• The set A is a subset of the set B

if every element of A is an element of B.

• If A is a subset of B and B contains elements which are not in A, then A is a proper subset of B.

August 2006 12

Notation - subsetIf A is a subset of B we write A B to designate that relationship.

If A is a proper subset of B we write A B to designate that relationship.

If A is not a subset of B we write A B to designate that relationship.

August 2006 13

Example - subset The set A = {1, 2, 3} is a subset of the set B

={1, 2, 3, 4, 5, 6} because each element of A is an element of B.

We write A B to designate this relationship between A and B.

We could also write {1, 2, 3} {1, 2, 3, 4, 5, 6}

August 2006 14

Example - subset The set A = {3, 5, 7} is not a subset

of the set B = {1, 4, 5, 7, 9} because 3 is an element of A but is not an element of B.

The empty set is a subset of every set, because every element of the empty set is an element of every other set.

August 2006 15

PROPER SUBSET:

• If A⊆B and A≠B then A is called a proper subset of B, written as A⊂B.

• For e.g.A={1,3,5},B={1,2,3,4,5,6}

August 2006 16

UNIVERSAL SET:• A set U is called a universal set. If all the sets

under consideration are sub-sets of the set U.• For e.g. If A {1,2,3,4},B {2,3,5,7} and

C={2,4,6,8}then the universal set U={1,2,3,4,5,6,7,8,9}.

POWER SET:• The collection of all possible subsets of a

given set A is the power set of A. It is denoted by P(A).

• For e.g. If A={1,2,3} then• P(A)={Ø,{1},{2},{3},{1,2},{2,3},

{1,3},{1,2,3}}.

August 2006 17

EQUAL SETS• Two sets A and B are equal if A

B and B A. If two sets A and B are equal we write A = B to designate that relationship.

August 2006 18

Example

The sets A = {3, 4, 6} and B = {6, 3, 4} are equal because A B and B A.The definition of equality of sets shows that the order in which elements are written does not affect the set.

August 2006 19

ExampleIf A = {1, 2, 3, 4, 5} and B = {x | x < 6 and x is a counting number} then A is a subset of B because every element of A is an element of B and B is a subset of A because every element of B is an element of A.

Therefore the two sets are equal and we write A = B.

August 2006 20

August 2006 21

Definition - union• The union of two sets A and B is the set

containing those elements which are elements of A or elements of B. We write A BIf A = {3, 4, 6} and B = { 1, 2, 3, 5, 6} then A B = {1, 2, 3, 4, 5, 6}.

August 2006 22

Example - UnionIf A is the set of prime numbers and B is the set of even numbers then A B = {x | x is even or x is prime }.

If A = {x | x > 5 } and B = {x | x < 3 } then A B = {x | x < 3 or x > 5 }.

August 2006 23

Venn Diagram - unionA is represented by the red circle and B isrepresented by the blue circle.The purple colored regionillustrates the intersection.The union consists of allpoints which are colored red or blue or purple.

A B

A∩B

August 2006 24

Definition - intersection• The intersection of two sets A

and B is the set containing those elements which are

elements of A and elements of B.

We write A B

August 2006 25

ExampleIf A = {3, 4, 6, 8} and B = { 1, 2, 3, 5, 6} then A B = {3, 6}If A is the set of prime numbers and B is the set of even numbers then A ∩ B = { 2 }

If A = {x | x > 5 } and B = {x | x < 3 } then A ∩ B =

August 2006 26

ExampleIf A = {x | x < 4 } and B = {x | x >1 } thenA ∩ B = {x | 1 < x < 4 }

If A = {x | x > 4 } and B = {x | x >7 } thenA ∩ B = {x | x < 7 }

August 2006 27

These sets can be visualized with circles in what is called a Venn Diagram.

A B

A BEverything that is in

A AND B.

August 2006 28

• Disjoint sets:Two sets A and B are disjoint sets, if A∩B=Ø.For e.g.

If A={2,4,6,8} and B={ 1,3,5,7} then A and B are disjoint sets.

Note: If A∩B ≠Ø, then A and B are said to be intersecting sets or overlapping sets.

• Difference of sets: If A and B are two sets, then their difference A-B is the set containing exactly those element in A that are not in B.

Eg:- A={a,b,c} B={a,d} A-B={b,c}

August 2006 29

• Symmetric difference:-Symmetric difference of two sets P&Q is the set containing exactly all the elements that are not in P or in Q but not in both.

P(+)Q=(PUQ)-(P ∩Q) Eg:-P={a,b} Q={a,c} Then symmetric difference is

{b,c}

August 2006 30

August 2006 31

• Union and intersection are commutative operations.

A B = B A A ∩ B = B ∩ A• Union and intersection are associative

operations. (A B) C = A (B C) (A ∩ B) ∩ C = B ∩ (A ∩ C)

August 2006 32

• Idempotent law:- AUA=A A∩A=A• Identity law:- AU=A AUU=U A∩U=A• Complement law:- AUA’=U U’=

August 2006 33

• Two distributive laws are true.

A ∩ ( B C )= (A ∩ B) (A ∩ C) A ( B ∩ C )= (A B) ∩ (A C)