chapter1 set edited

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Chapter 1:

Discrete Structures

SETS



SETS

• WHY ARE WE STUDYING SETS

• The concept of set is basic to all of

mathematics and mathematical applications.

• Sets are foundational in many areas ofComputer Science.

• For us, sets are useful to understand the

principles of counting and probability theory



SETS

A set is determined by its elements and

not by any particular order in which the

element might be listed.Example,

A={1, 2, 3, 4},

A might just as well be specified as

{2, 3, 4, 1} or {4, 1, 3, 2}



SETS

The elements making up a set are

assumed to be distinct, we may have

duplicates in our list, only one occurrenceof each element is in the set.

Example

{a, b, c, a, c} = {a, b, c}

{1, 3, 3, 5, 1} = {1, 3, 5}



SETS

Use uppercase letters A, B, C … to

denote setsThe symbol ∈ stands for “belongs to”

The symbol ∉ stands for “does not

belong to”



EXAMPLE

X={ a, b, c, d, e },

b∈ X and m∉ X

A={{1}, {2}, 3, 4},

{2}∈ A and 1∉ A



SETS

If a set is a large finite set or an infinite set,we can describe it by listing a property

necessary for memberships

Let S be a set, the notation, A= {x | x ∈ S, P(x)} or A= {x ∈ S | P(x)}

means that A is the set of all elements x of

S such that x satisfies the property P.



SETS

Let A={1, 2, 3, 4, 5, 6}, we can also

write A as,

A={x | x ∈ Z, 0<x<7}

if Z denotes the set of integers.

Let B={x | x ∈ Z, x>0},B={1, 2, 3, 4, …}



SUBSET

If every element of A is an element

of B, we say that A is a subset of B

and write A ⊆ B.

A=B, if A ⊆ B and B ⊆ A.

The empty set { } is a subset of every

set.



EXAMPLE

A={1, 2, 3}

Subset of A,

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



PROPER SUBSET

If A is a subset of B and A is

not equal to B, we say that A

is a proper subset of B.

A⊂ B and A≠B



EXAMPLE

B={1, 2, 3}

Proper subset of B,

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



EQUAL SET

Two set A and B are equal and we write

A=B, if A and B have the same elements.

example A={a, b, c}, B={b, c, a}, A=B

C={1, 2, 3, 4}D={x | x is a positive integer , 2x <10}

C=D



UNIVERSAL SET

Sometimes we are dealing with sets all

of which are subsets of a set U.

This set U is called a universal set or auniverse.

The set U must be explicitly given or

inferred from the context.



EXAMPLE

The sets X={1,2,3}, Y={2,4,6,8} and Z={5,7}

One may choose U={1,2,3,4,5,6,7,8} as auniversal set.



Finite Set

Let A be a set,

If there exists a nonnegative integer

n such that A has n elements, then Ais called a finite set with n elements.

Example

C = {1, 2, 3, 4}

B = {x | x is an integer, 1 < x < 4}



Infinite SET

Let A be a set, A is called an infinite set, if A is

not a finite set.

Example Z = {x| x is an integer}

or Z = {…, -3, -2, -1, 0, 1, 2, 3,…}

S={x| x is a real number , 1 < x < 4}



CARDINALITY SET

Let S be a finite set with n distinct

elements, where n≥0.

Then we write |S|=n and say that thecardinality (or the number of elements) of

S is n.

example A= {1, 2, 3}, |A|=3

B= {a,b,c,d,e,f,g}, |B|=7



POWER SET

The set of all subsets of a set A,

denoted

P(A), is called the power set of A.P(A)= {X | X ⊆ A}

If | A|=n, then |P(A)| = 2^n



EXAMPLE

A={1,2,3}

the power set of A,

P(A)= {{ }, {1}, {2}, {3}, {1,2}, {1,3}, {2,3},

{1,2,3} }Notice that| A| = 3, and|P(A)| = 2^3 = 8



EXERCISE

Let X= {1, 2, 2, {1}, a}

Find:

| X|

Proper subset of X

Power set of X



OPERATION ON SET

The union of two sets A and B, denoted by A

∪ B, is defined to be the set

A ∪ B = { x | x ∈ A or x ∈ B}

The union consists of all elements belongingto either A or B (or both)



OPERATIONS ON SETS

Venn diagram of A ∪ B



EXAMPLE

A={1, 2, 3, 4, 5}, B={2, 4, 6} and C={8, 9}

Find : A ∪ B =

A ∪ C =

B ∪ C =

A ∪ B ∪ C =



A ∪ B = {1, 2, 3, 4, 5, 6}

A ∪ C = {1, 2, 3, 4, 5, 8, 9}

B ∪ C = {2, 4, 6, 8, 9}

A ∪ B ∪ C = {1, 2, 3, 4, 5, 6, 8, 9}



OPERATIONS ON SETS

The intersection of two sets A and B,

denoted by A ∩ B, is defined to be the

set

A ∩ B = { x | x ∈ A and x ∈ B}

The intersection consists of all

elements belonging to both A and B.



OPERATIONS ON SETS

Venn diagram of A ∩ B



EXAMPLES

A={1, 2, 3, 4, 5, 6},

B={2, 4, 6, 8, 10} and

C={ 1, 2, 8, 10 }

Find: A ∩ B =

A ∩ C =

C ∩ B = A ∩ B ∩ C =



A ∩ B = {2, 4, 6}

A ∩ C = {1, 2}

C ∩ B = {2, 8, 10}

A ∩ B ∩ C = {2}



OPERATIONS ON SETS

Venn diagram, A ∩ B = ∅



OPERATIONS ON SETS

If A and B are finite sets, the

cardinality of A ∪ B,

| A ∪ B| = |A| + |B| − |A ∩ B|



OPERATIONS ON SETS

The set

A-B= { x | x ∈ A and x ∉ B}

is called the difference.

The difference A-B consists of all

elements in A that are not in B.



OPERATIONS ON SETS

Venn diagram of A-B



EXAMPLE

A= { 1, 2, 3, 4, 5, 6, 7, 8 }

B= { 2, 4, 6, 8 }

A-B = { 1, 3, 5, 7 }



OPERATIONS ON SETS

The complement of a set A with respect to

a universal set U, denoted by A′ is defined

to be

A′ = { x ∈ U| x ∉ A}

A′ = U-A



OPERATIONS ON SETS

Venn diagram of A’



EXAMPLE

Let U be a universal set,

U= { 1, 2, 3, 4, 5, 6, 7 }

A= { 2, 4, 6 }

A′ = U – A = { 1, 3, 5, 7 }



EXERCISE

Let,

U = { a, b, c, d, e, f , g, h, i, j, k, l, m }

A = { a, c, f, m}

B = { b, c, g, h, m }

Find:

A ∪ B , A ∩ B , | A ∪ B| , A-B and A′.



SOLUTIONS

A ∪ B = { a,b,c,f,g,h,m }

A ∩ B = { c,m }

| A ∪ B| = 7

A-B = { a,f }

A′ = { b,d,e,g,h,i,j,k,l}



{1,2,3} ∪ {3,4,5}

{x| x>0} ∪ {x| x>1}

{1,2,3} ∩ {3,4,5

{x| x>0} ∩ {x| x>1}

EXERCISE



{1,2,3}∩ {3,4,5}={3}

{x| x>0}∩ {x| x>1} = {x| x>1}

{1,2,3}∪ {3,4,5}={1,2,3,4,5}

{x| x>0}∪ {x| x>1} = {x| x>0}

SOLUTIONS

P ti f S t



Properties of Set

• Commutative laws

• A ∩ B =B ∩ A

• A ∪ B =B ∪ A



Properties of Set

• Assoc ia tive laws

• A ∩ (B ∩ C ) = (A ∩ B ) ∩ C

• A ∪ (B ∪ C ) = (A ∪ B ) ∪ C

• Distributive laws

• A ∪ (B ∩ C ) = (A ∪ B ) ∩ (A ∪ C )• A ∩ (B ∪ C ) = (A ∩ B ) ∪ (A ∩ C )



Properties of Set

• De Morgan’s laws

• (A ∩ B )′ = A′ ∪ B ′

• (A ∪ B )′ = A′ ∩ B ′

• Properties of universa l set

• A ∪ U = U • A ∩ U = A



Properties of SET

• Properties of empty set

• A ∪ ∅ = A

• A ∩ ∅ = ∅



EXAMPLE

Simplify the set

(((A∪B)∩C)’∪B’)’

SOLUTION



SOLUTION



Cartesian Produc t

• An ordered pa ir (a , b ) is c onsidered

d istinc t from ordered pa ir (b , a ), unless

a =b .

• Examp le (1, 2) ≠ (2, 1)



Cartesian Produc t

• The Cartesian product of two sets A

and B, written A× B is the set,

• A× B = {(a,b)| a∈ A, b∈ B}• For any set A,

• A×∅ = ∅× A = ∅



EXAMPLE

• A= {a , b }, B ={1, 2}.

• A×B = {(a , 1), (a , 2), (b , 1), (b , 2)}

• B ×A = {(1, a ), (1, b ), (2, a ), (2, b )}



Cartesian Produc t

• if A ≠ B, then A× B ≠ B× A.

• if | A| = m and | B| = n, then | A× B|=mn



EXAMPLE

• A= {1, 3}, B ={2, 4, 6}.

• A×B = {(1, 2), (1, 4), (1, 6), (3, 2), (3, 4),

(3, 6)}

• B ×A = {(2, 1), (2, 3), (4, 1), (4, 3), (6, 1),

(6, 3)}

• A ≠ B , A×B ≠ B ×A

• | A| = 2 , | B | = 3, | A×B | = 2.3= 6.



EXERCISE

• {1,2 } X {3, 4,5}

• {Male, Female} X {Married, Single} X

{ Student, Faculty}



Exercise

• Let A= {w , x }, B={1, 2} and C ={KB, SD,PS}.

• Find | A×B | , | B ×C| ,

| A×C| ,| A×B ×C|

• Determine the following set,

• A×B, B ×C, A×C

• A×B ×C

chapter1 set edited

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