Discrete MathematicsDiscrete Mathematics
and Itsand Its
ApplicationsApplications
Seventh EditionSeventh Edition
Chapter 2Chapter 2
SetsSets
Lecture Slides By Adil AslamLecture Slides By Adil Aslam
Lecture Slides By Adil Aslam
Email Address is : adilaslam5959@gmail.com

SetsSets
German mathematician German mathematician G. CantorG. Cantor introduced introduced
the concept of sets. He had defined a setthe concept of sets. He had defined a set
as a collection of definite andas a collection of definite and
distinguishable objects selected by thedistinguishable objects selected by the
means of certain rules or description.means of certain rules or description.
SetSet theory forms the basis of several other theory forms the basis of several other
fields of study like counting theory,fields of study like counting theory,
relations, graph theory and finite staterelations, graph theory and finite state
machines. In this chapter, we will cover themachines. In this chapter, we will cover the
different aspects of different aspects of Set TheorySet Theory..
Lecture Slides By Adil Aslam

Set TheorySet Theory
A well defined collection of {distinct} objectsA well defined collection of {distinct} objects
is called a set.is called a set.
– The objects are called the elements orThe objects are called the elements or
members of the set.members of the set.
– Sets are denoted by capital lettersSets are denoted by capital letters
A, B, C …, X, Y, Z.A, B, C …, X, Y, Z.
– The elements of a set are represented by lowerThe elements of a set are represented by lower
case letterscase letters
a, b, c, … , x, y, z.a, b, c, … , x, y, z.
Lecture Slides By Adil Aslam

Set TheorySet Theory
• Set: Collection of objects (“elements”)Set: Collection of objects (“elements”)
• aa∈∈AA “a is an element of A”“a is an element of A”
“a is a member of A”“a is a member of A”
• aa∉∉AA “a is not an element of A”“a is not an element of A”
• A = {aA = {a11, a, a22, …, a, …, ann}} “A contains…”“A contains…”
• Order of elements is meaninglessOrder of elements is meaningless
• It does not matter how often the sameIt does not matter how often the same
element is listed.element is listed.
Lecture Slides By Adil Aslam

Set TheorySet Theory
A set is an unordered collection of different
elements. A set can be written explicitly by
listing its elements using set bracket. If
the order of the elements is changed or
any element of a set is repeated, it does
not make any changes in the set.
Some Example of Sets
A set of all positive integers
A set of all the planets in the solar system
A set of all the states in India
A set of all the lowercase letters of the
alphabet Lecture Slides By Adil Aslam

SetsSets
Definition:Definition:
 A set is an unordered collection of objectsA set is an unordered collection of objects
referred to as elements.referred to as elements.
 A set is said to contain its elements.A set is said to contain its elements.
 We write a A to denote that a is an∈We write a A to denote that a is an∈
element of the set A.element of the set A.
 The notation a ~ A denotes that a is not∈The notation a ~ A denotes that a is not∈
an element of the set A.an element of the set A.
Lecture Slides By Adil Aslam

Examples of SetExamples of Set
 EXAMPLE 1:EXAMPLE 1:
The set V of all vowels in the EnglishThe set V of all vowels in the English
alphabet can be written as V={a, e, i, o, u}.alphabet can be written as V={a, e, i, o, u}.
 EXAMPLE 2:EXAMPLE 2:
The set O of odd positive integers less thanThe set O of odd positive integers less than
10 can be expressed by O={1,3,5,7,9}.10 can be expressed by O={1,3,5,7,9}.
Lecture Slides By Adil Aslam

Representation of a SetRepresentation of a Set
Sets can be represented in two
ways :
• Roster or Tabular Form
• Descriptive Form
• Set Builder Notation
Lecture Slides By Adil Aslam

Set TheorySet Theory
TABULAR FORMTABULAR FORM
– Listing all the elements of a set, separated byListing all the elements of a set, separated by
commas and enclosed within braces or curlycommas and enclosed within braces or curly
brackets{}.brackets{}.
EXAMPLESEXAMPLES
– In the following examples we write the sets inIn the following examples we write the sets in
Tabular Form.Tabular Form.
– A = {1, 2, 3, 4, 5}A = {1, 2, 3, 4, 5} is the set of first fiveis the set of first five
Natural NumbersNatural Numbers..
– B = {2, 4, 6, 8, …, 50} is the set ofB = {2, 4, 6, 8, …, 50} is the set of EvenEven
numbersnumbers up to 50up to 50
– C = {1, 3, 5, 7, 9, …} is the set ofC = {1, 3, 5, 7, 9, …} is the set of positive oddpositive odd
numbers.numbers.
Lecture Slides By Adil Aslam

Set TheorySet Theory
Descriptive Form:Descriptive Form:
– Stating in words the elements of a set.Stating in words the elements of a set.
EXAMPLESEXAMPLES
– Now we will write the same examples which weNow we will write the same examples which we
write in Tabular Form ,in the Descriptive Form.write in Tabular Form ,in the Descriptive Form.
– A = set of first five Natural Numbers.( is theA = set of first five Natural Numbers.( is the
Descriptive Form )Descriptive Form )
– B = set of positive even integers less or equal toB = set of positive even integers less or equal to
fifty. ( is the Descriptive Form )fifty. ( is the Descriptive Form )
– C = {1, 3, 5, 7, 9, …}C = {1, 3, 5, 7, 9, …} ( is the Tabular Form )( is the Tabular Form )
– C = set of positive odd integers.C = set of positive odd integers. ( is the( is the
Descriptive Form )Descriptive Form )
Lecture Slides By Adil Aslam

Set TheorySet Theory
Set Builder FormSet Builder Form
– Writing in symbolic form the commonWriting in symbolic form the common
characteristics shared by all the elements of thecharacteristics shared by all the elements of the
set.set.
EXAMPLESEXAMPLES
– Now we will write the same examples which weNow we will write the same examples which we
write in Tabular as well as Descriptive Form ,inwrite in Tabular as well as Descriptive Form ,in
Set Builder Form .Set Builder Form .
– A = {xA = {x ∈∈NN || x<=5} ( is the Set Builder Form)x<=5} ( is the Set Builder Form)
– B = {xB = {x ∈∈ EE || 0 < x <=50} ( is the Set Builder Form)0 < x <=50} ( is the Set Builder Form)
– C = {xC = {x ∈∈OO || 0 < x } ( is the Set Builder Form)0 < x } ( is the Set Builder Form)
Lecture Slides By Adil Aslam

SymbolSymbol
{ } Represent the set{ } Represent the set
| Such that| Such that
∈∈ Is a member ofIs a member of
ExampleExample
{x | x < 5}{x | x < 5}
The set of all real numbers less than 5The set of all real numbers less than 5
{ k member of real's | k > 5 }{ k member of real's | k > 5 }
The set of all k's that are a member of theThe set of all k's that are a member of the
Integers, such that k is greater than 5"Integers, such that k is greater than 5"
Lecture Slides By Adil Aslam

Famous Sets in MathFamous Sets in Math
NN =Set of natural numbers=Set of natural numbers
ZZ =Set of integers.=Set of integers.
Z+Z+ = Set of positive integers.= Set of positive integers.
QQ = {p/q | p Z, q Z, and q≠0}, set of∈ ∈= {p/q | p Z, q Z, and q≠0}, set of∈ ∈
rational numbers.rational numbers.
Q+Q+ =The set of positive rational number.=The set of positive rational number.
RR = The set of real numbers.= The set of real numbers.
R+R+ =The set of positive real numbers.=The set of positive real numbers.
CC =The set of complex numbers=The set of complex numbers
Lecture Slides By Adil Aslam

Examples for SetsExamples for Sets
““Standard” Sets:Standard” Sets:
• Natural numbersNatural numbers NN = {0, 1, 2, 3, …}= {0, 1, 2, 3, …}
• IntegersIntegers ZZ = {…, -2, -1, 0, 1, 2, …}= {…, -2, -1, 0, 1, 2, …}
• Positive IntegersPositive Integers ZZ++
= {1, 2, 3, 4, …}= {1, 2, 3, 4, …}
• Real NumbersReal Numbers RR = {47.3, -12,= {47.3, -12, ππ, …}, …}
• Rational NumbersRational Numbers QQ = {1.5, 2.6, -3.8, 15, …}= {1.5, 2.6, -3.8, 15, …}
NOTE:NOTE:
– The symbol “…” is called an ellipsis. It is a shortThe symbol “…” is called an ellipsis. It is a short
for “and so forth.”for “and so forth.”
Lecture Slides By Adil Aslam

SetsSets
 When a and b are real numbers withWhen a and b are real numbers with
a<b, we writea<b, we write
[a, b]={x|a≤x≤b}[a, b]={x|a≤x≤b}
[a, b)={x|a≤x<b}[a, b)={x|a≤x<b}
(a, b]={x|a<x≤b}(a, b]={x|a<x≤b}
(a, b)={x|a<x<b}(a, b)={x|a<x<b}
Note that [a, b] is called the closed intervalNote that [a, b] is called the closed interval
from a to b.from a to b.
(a, b) is called the open interval from a to b.(a, b) is called the open interval from a to b.
(a, b]is called half-open interval from a to b.(a, b]is called half-open interval from a to b.
Lecture Slides By Adil Aslam

Equal SetsEqual Sets
 Two sets are equal if and only if they haveTwo sets are equal if and only if they have
the same elements.the same elements.
 If A and B are sets, then A and B areIf A and B are sets, then A and B are
equal if and only if x(x A x B).∀ ∈ ↔ ∈equal if and only if x(x A x B).∀ ∈ ↔ ∈
 We write A=B if A and B are equal sets.We write A=B if A and B are equal sets.
 ExampleExample
 A={1,5,7} and B={7,5,1}are equal, becauseA={1,5,7} and B={7,5,1}are equal, because
they have the same elements.they have the same elements.
Lecture Slides By Adil Aslam

ExampleExample
For Example;For Example;
A = {2, 3, 5} A = {2, 3, 5}
B = {5, 2, 3}B = {5, 2, 3}
Here, set A and set B are equal setsHere, set A and set B are equal sets
Lecture Slides By Adil Aslam

Your taskYour task
LetLet
A = {1, 2, 3, 6}A = {1, 2, 3, 6} B = the set of positiveB = the set of positive
divisors of 6divisors of 6
C = {3, 1, 6, 2}C = {3, 1, 6, 2} D = {1, 2, 2, 3, 6, 6, 6}D = {1, 2, 2, 3, 6, 6, 6}
Is set A , B , C , D are Equal sets?Is set A , B , C , D are Equal sets?
Solution:Solution:
Yes A, B, C, and D are all equal sets.Yes A, B, C, and D are all equal sets.
Lecture Slides By Adil Aslam

Other TaskOther Task
Is this is equal set?Is this is equal set?
{2,3,5,7} , {2,2,3,5,3,7}{2,3,5,7} , {2,2,3,5,3,7}
EqualEqual
AndAnd
{2,3,5,7} , {2,3}{2,3,5,7} , {2,3}
Not EqualNot Equal
Lecture Slides By Adil Aslam

Equivalent SetsEquivalent Sets
 Two finite sets A and B are said toTwo finite sets A and B are said to
be equivalent (be equivalent (written A =~ Bwritten A =~ B) if they) if they
have the same number of elements:have the same number of elements:
that is, n(A) = n(B).that is, n(A) = n(B).
 ExampleExample
 {p, q, r, s}; {a, b, c, d}{p, q, r, s}; {a, b, c, d}
Lecture Slides By Adil Aslam

ExamplesExamples
For Example;For Example;
A = {p, q, r} A = {p, q, r}
B = {2, 3, 4} B = {2, 3, 4}
Here, we observe that both the setsHere, we observe that both the sets
contain three elements.contain three elements.
Example: If A = {1, 2, 6} and B = {16,
17, 22}, they are equivalent
Lecture Slides By Adil Aslam

Note thatNote that
Equal sets are always equivalent. Equal sets are always equivalent.
Equivalent sets may not be equalEquivalent sets may not be equal
Lecture Slides By Adil Aslam

Empty Set or NULL SetEmpty Set or NULL Set
 The empty set is a set which has noThe empty set is a set which has no
elements.elements.
 It is also called null set.It is also called null set.
 It is denoted byIt is denoted by ∅∅ or by {}.or by {}.
Note the subtlety inNote the subtlety in ∅∅ ≠≠ {{∅∅}}
– The left-hand side is the empty setThe left-hand side is the empty set
– The right hand-side is a singleton set, and a setThe right hand-side is a singleton set, and a set
containing a setcontaining a set
Lecture Slides By Adil Aslam

Singleton Set or Unit SetSingleton Set or Unit Set
 A singleton is a set that containsA singleton is a set that contains
exactly one element.exactly one element.
 Sometimes, it is known as unit set.Sometimes, it is known as unit set.
 The singleton containing only theThe singleton containing only the
element a can be written {a}.element a can be written {a}.
 Note thatNote that фф is empty set and {is empty set and {фф} is} is
not empty set but it is a singletonnot empty set but it is a singleton
set.set.
Lecture Slides By Adil Aslam

Singleton Set or Unit SetSingleton Set or Unit Set
Singleton set or unit set contains onlySingleton set or unit set contains only
one element. A singleton set is denotedone element. A singleton set is denoted
by {s}.by {s}.
ExampleExample : : S = {x | x N, 7 < x < 9}∈S = {x | x N, 7 < x < 9}∈
Lecture Slides By Adil Aslam

Universal SetUniversal Set
 A Universal Set is the set of all elementsA Universal Set is the set of all elements
under consideration, is represented by aunder consideration, is represented by a
rectangle.rectangle.
 It is denoted by capital U.It is denoted by capital U.
 Note that the universal set variesNote that the universal set varies
depending on which objects are ofdepending on which objects are of
interest.interest.
 Inside this rectangle, circles or otherInside this rectangle, circles or other
geometrical figures are used to representgeometrical figures are used to represent
sets.sets. Lecture Slides By Adil Aslam

Venn DiagramsVenn Diagrams
Venn diagram, invented in 1880 by John
Venn, is a schematic diagram that
shows all possible logical relations
between different mathematical sets.
Lecture Slides By Adil Aslam

Venn DiagramsVenn Diagrams
 Venn diagrams are often used to indicateVenn diagrams are often used to indicate
the relationships between sets.the relationships between sets.
 ExampleExample
Venn diagram that represents V, the set ofVenn diagram that represents V, the set of
vowels in the English alphabetvowels in the English alphabet..
Lecture Slides By Adil Aslam

SubsetSubset
 The set A is a subset of B if and only ifThe set A is a subset of B if and only if
every element of A is also an element of B.every element of A is also an element of B.
 We use the notation A B to indicate that⊆We use the notation A B to indicate that⊆
A is a subset of the set B.A is a subset of the set B.
 The empty set( { } orThe empty set( { } or фф ) is a subset of) is a subset of
every set.every set.
 ExampleExample
 A = {1,2,3,4,7}A = {1,2,3,4,7}
 B = {1,2,3,4,7,8}B = {1,2,3,4,7,8}
 Here, A is said to be the subset.Here, A is said to be the subset.
Lecture Slides By Adil Aslam

SubsetsSubsets
AA ⊆⊆ BB “A is a subset of B”“A is a subset of B”
AA ⊆⊆ B if and only if every element of A is alsoB if and only if every element of A is also
an element of B.an element of B.
We can completely formalize this:We can completely formalize this:
AA ⊆⊆ BB ⇔⇔ ∀∀x (xx (x∈∈AA →→ xx∈∈B)B)
Examples:Examples:
A = {3, 9}, B = {5, 9, 1, 3}, AA = {3, 9}, B = {5, 9, 1, 3}, A  B ?B ? truetrue
A = {3, 3, 3, 9}, B = {5, 9, 1, 3}, AA = {3, 3, 3, 9}, B = {5, 9, 1, 3}, A  B ?B ?
falsefalse
truetrue
A = {1, 2, 3}, B = {2, 3, 4}, AA = {1, 2, 3}, B = {2, 3, 4}, A  B ?B ?
Lecture Slides By Adil Aslam

Proper SubsetProper Subset
 A set B is said to be the proper subset ofA set B is said to be the proper subset of
any given set A, if B has some elementsany given set A, if B has some elements
form the set A but B is not equal to theform the set A but B is not equal to the
set A (B≠ A).set A (B≠ A).
 ExampleExample
 A = {1, 2, 3,4 ,5, 6}A = {1, 2, 3,4 ,5, 6}
 B = { 1,2,3,4}B = { 1,2,3,4}
 B is proper subset of the set A.(B A)⊂B is proper subset of the set A.(B A)⊂
Lecture Slides By Adil Aslam

Your TaskYour Task
Determine whether each of the followingDetermine whether each of the following
statements is true or false.statements is true or false.
– xx ∈∈ {x}{x}
– {x}{x}⊆⊆ {x}{x}
– {x}{x} ∈∈{x}{x}
– {x}{x} ∈∈{{x}}{{x}}
∅∅ ⊆⊆ {x}{x}
∅∅ ∈∈ {x}{x}
Lecture Slides By Adil Aslam

SolutionSolution
Determine whether each of the followingDetermine whether each of the following
statements is true or false.statements is true or false.
– xx ∈∈ {x}{x} TRUETRUE
• ( Because x is the member of the singleton set { x } )( Because x is the member of the singleton set { x } )
– {x}{x}⊆⊆ {x}{x} TRUETRUE
• (Because Every set is the subset of itself. Note that(Because Every set is the subset of itself. Note that
every Set has necessarily two subsetsevery Set has necessarily two subsets ∅∅ and the Setand the Set
itself, these two subset are known as Improper subsetsitself, these two subset are known as Improper subsets
and any other subset is called Proper Subset)and any other subset is called Proper Subset)
– {x}{x} ∈∈{x}{x} FALSEFALSE
• ( Because { x} is not the member of {x} ) Similarly other( Because { x} is not the member of {x} ) Similarly other
– {x}{x} ∈∈{{x}}{{x}} TRUETRUE
∅∅ ⊆⊆ {x}{x} TRUETRUE
∅∅ ∈∈ {x}{x} FALSEFALSE
Lecture Slides By Adil Aslam

Set TheorySet Theory
Finite SetsFinite Sets
– A setA set SS is said to beis said to be finitefinite if it contains exactlyif it contains exactly mm
distinct elements wheredistinct elements where mm denotes some nondenotes some non
negative integer.negative integer.
– In such case we writeIn such case we write ||SS|| == mm or n(S) =or n(S) = mm
Infinite SetsInfinite Sets
– A set is said to beA set is said to be infiniteinfinite if it is not finite.if it is not finite.
ExamplesExamples
– The set S of letters of English alphabets is finiteThe set S of letters of English alphabets is finite
andand ||SS|| = 26= 26
– The null setThe null set ∅∅ has no elements, is finite andhas no elements, is finite and |∅||∅| = 0= 0
– The set of positive integers {1, 2, 3,…} is infinite.The set of positive integers {1, 2, 3,…} is infinite.
Lecture Slides By Adil Aslam

Your TaskYour Task
Determine which of the following sets areDetermine which of the following sets are
finite/infinite.finite/infinite.
– A = {month in the year}A = {month in the year} FINITEFINITE
– B = {even integers}B = {even integers} INFINITEINFINITE
– C = {positive integers less than 1}C = {positive integers less than 1} FINITEFINITE
– D = {animals living on the earth}D = {animals living on the earth} FINITEFINITE
– E = {lines parallel to x-axis}E = {lines parallel to x-axis} INFINITEINFINITE
Lecture Slides By Adil Aslam

Cardinality of SetsCardinality of Sets
 The cardinality of a set S, denoted |S|, isThe cardinality of a set S, denoted |S|, is
the number of elements in S. If the setthe number of elements in S. If the set
has an infinite number of elements, thenhas an infinite number of elements, then
its cardinality is ∞.its cardinality is ∞.
 The cardinality of a set A is denoted byThe cardinality of a set A is denoted by
| A |.| A |.
 1: If A =1: If A = фф , then | A |= 0., then | A |= 0.
 2: If A has exactly n elements, then2: If A has exactly n elements, then
| A | = n.| A | = n.
 Note that n is a nonnegative number.Note that n is a nonnegative number.
 3: If A is an infinite set, then | A | = ∞.3: If A is an infinite set, then | A | = ∞.
Lecture Slides By Adil Aslam

ExamplesExamples
 Let A be the set of odd positiveLet A be the set of odd positive
integers less than 10. Thenintegers less than 10. Then | A |=5| A |=5..
 Let S be the set of letters in theLet S be the set of letters in the
English alphabet. ThenEnglish alphabet. Then | S |=26| S |=26 ..
 Let P be the set of infinite numbers.Let P be the set of infinite numbers.
ThenThen | P |=∞| P |=∞..
– The cardinality of the empty set is |The cardinality of the empty set is |∅∅||
=0=0
– The setsThe sets NN,, ZZ,, QQ,, RR are all infiniteare all infinite
Lecture Slides By Adil Aslam

Cardinality of SetsCardinality of Sets
Your Task!Your Task!
Examples:Examples:
A = {Mercedes, BMW, Porsche}, |A| = 3A = {Mercedes, BMW, Porsche}, |A| = 3
B = {1, {2, 3}, {4, 5}, 6}B = {1, {2, 3}, {4, 5}, 6} |B| = 4|B| = 4
C = C =  |C| = 0|C| = 0
D = { xD = { xNN | x  7000 }| x  7000 } |D| = 7001|D| = 7001
E = { xE = { xNN | x  7000 }| x  7000 } E is infinite!E is infinite!
Lecture Slides By Adil Aslam

Power SetsPower Sets
 A Power Set is a set of all theA Power Set is a set of all the
subsets of a set.subsets of a set.
 The power set of S is denoted byThe power set of S is denoted by
P(S).P(S).
 NotationNotation
 The number of members of a set isThe number of members of a set is
often written as |S|, so we can writeoften written as |S|, so we can write
|P(S)| = 2|P(S)| = 2nn
Lecture Slides By Adil Aslam

ExamplesExamples
 A= { a,b,c,d}A= { a,b,c,d}
The power set of A is 2 = 16⁴The power set of A is 2 = 16⁴
P(A)={}, {a}, {b}, {a, b}, {c}, {a, c}, {b, c}, {a, b,P(A)={}, {a}, {b}, {a, b}, {c}, {a, c}, {b, c}, {a, b,
c}, {d}, {a, d}, {b, d}, {a, b, d}, {c, d}, {a, c,c}, {d}, {a, d}, {b, d}, {a, b, d}, {c, d}, {a, c,
d}, {b, c, d}, {a, b, c, d}.d}, {b, c, d}, {a, b, c, d}.
 B={1,2,3}B={1,2,3}
The power set of B is 2³=8The power set of B is 2³=8
P(B)={}, {1}, {2}, {1, 2}, {3}, {1, 3}, {2, 3}, {1, 2,P(B)={}, {1}, {2}, {1, 2}, {3}, {1, 3}, {2, 3}, {1, 2,
3}3}
Lecture Slides By Adil Aslam

Your Task NowYour Task Now
 C={a,1,b,2,c}C={a,1,b,2,c}
P(C)={}, {a}, {1}, {a, 1}, {b}, {a, b}, {1, b},P(C)={}, {a}, {1}, {a, 1}, {b}, {a, b}, {1, b},
{a, 1, b}, {2}, {a, 2}, {1, 2}, {a, 1, 2}, {b,{a, 1, b}, {2}, {a, 2}, {1, 2}, {a, 1, 2}, {b,
2}, {a, b, 2}, {1, b, 2}, {a, 1, b, 2}, {c},2}, {a, b, 2}, {1, b, 2}, {a, 1, b, 2}, {c},
{a, c}, {1, c}, {a, 1, c}, {b, c}, {a, b, c},{a, c}, {1, c}, {a, 1, c}, {b, c}, {a, b, c},
{1, b, c}, {a, 1, b, c}, {2, c}, {a, 2, c}, {1,{1, b, c}, {a, 1, b, c}, {2, c}, {a, 2, c}, {1,
2, c}, {a, 1, 2, c}, {b, 2, c}, {a, b, 2, c},2, c}, {a, 1, 2, c}, {b, 2, c}, {a, b, 2, c},
{1, b, 2, c}, {a, 1, b, 2, c}.{1, b, 2, c}, {a, 1, b, 2, c}.
Lecture Slides By Adil Aslam

Your TaskYour Task
a.a. FindFind PP((∅∅))
b.b. FindFind PP((PP((∅∅))))
c.c. FindFind PP((PP((PP((∅∅))))))
Solution:Solution:
– SinceSince ∅∅ contains no element, thereforecontains no element, therefore
PP((∅∅) will contain 2) will contain 200
=1 element.=1 element.
PP((∅∅) = {) = {∅∅}}
– SinceSince PP((∅∅) contains one element, namely) contains one element, namely φφ,,
thereforetherefore PP((∅∅) will contain 2) will contain 211
= 2 elements= 2 elements
PP((PP((∅∅)) = {)) = {∅∅,{,{∅∅}}}}
Lecture Slides By Adil Aslam

Cartesian ProductsCartesian Products
 Let A and B be sets. The CartesianLet A and B be sets. The Cartesian
product of A and B, denoted by A×B, is theproduct of A and B, denoted by A×B, is the
set of all ordered pairsset of all ordered pairs
 (a, b), where a A and b B.∈ ∈(a, b), where a A and b B.∈ ∈
 2-tuples are called ordered pairs.2-tuples are called ordered pairs.
 The ordered pairs (a, b)and(c, d)are equalThe ordered pairs (a, b)and(c, d)are equal
if and only if a=c and b=d. Note that (a, b)if and only if a=c and b=d. Note that (a, b)
and (b, a) are not equal unless a=b.and (b, a) are not equal unless a=b.
Lecture Slides By Adil Aslam

ExamplesExamples
 Cartesian product of set A and B is notCartesian product of set A and B is not
equal to Cartesian product of set B and A.equal to Cartesian product of set B and A.
 Cartesian product of sets “A” and “B” isCartesian product of sets “A” and “B” is
denoted by A×B .denoted by A×B .
 And Cartesian product of sets “B” and “A”And Cartesian product of sets “B” and “A”
is denoted by B×A.is denoted by B×A.
 ExampleExample
 A={1,2} and set B={4,5}A={1,2} and set B={4,5}
 A×B=[ {1,4} , {1,5} , {2,4} , {2,5} ]A×B=[ {1,4} , {1,5} , {2,4} , {2,5} ]
 B×A=[ {4,1} , {4,2} , {5,1} , {5,2} ]B×A=[ {4,1} , {4,2} , {5,1} , {5,2} ]
Lecture Slides By Adil Aslam

Cartesian ProductCartesian Product
TheThe Cartesian productCartesian product of two sets is defined as:of two sets is defined as:
AA××B = {(a, b) | aB = {(a, b) | a∈∈AA ∧∧ bb∈∈B}B}
Example:Example:
A = {good, bad}, B = {student, prof}A = {good, bad}, B = {student, prof}
AA××B = {B = {(good, student),(good, student), (good, prof),(good, prof), (bad, student),(bad, student), (bad, prof)(bad, prof)}}
(student, good),(student, good), (prof, good),(prof, good), (student, bad),(student, bad), (prof, bad)(prof, bad)}}BA = {BA = {
Lecture Slides By Adil Aslam

ExamplesExamples
 What is the Cartesian product A×B×C,What is the Cartesian product A×B×C,
where A={0,1} B={1,2}, and C={0,1,2}?where A={0,1} B={1,2}, and C={0,1,2}?
Ans:Ans:
A×B×C={(0,1,0), (0,1,1), (0,1,2), (0,2,0), (0,2,1),A×B×C={(0,1,0), (0,1,1), (0,1,2), (0,2,0), (0,2,1),
(0,2,2), (1,1,0), (1,1,1),(1,1,2), (1,2,0), (1,2,1),(0,2,2), (1,1,0), (1,1,1),(1,1,2), (1,2,0), (1,2,1),
(1,2,2)}.(1,2,2)}.
 Let A={1,2} and B= {3,4}Let A={1,2} and B= {3,4}
A×B= {(1,3), (1,4), (2,3), (2,4)}.A×B= {(1,3), (1,4), (2,3), (2,4)}.
Lecture Slides By Adil Aslam

Cartesian ProductCartesian Product
Note that:Note that:
• AA×∅×∅ == ∅∅
• ∅×∅×A =A = ∅∅
• For non-empty sets A and B: AFor non-empty sets A and B: A≠≠BB ⇔⇔ AA××BB ≠≠ BB××AA
• |A|A××B| = |A|B| = |A|⋅⋅|B||B|
The Cartesian product ofThe Cartesian product of two or more setstwo or more sets isis
defined as:defined as:
AA11××AA22××……××AAnn = {(a= {(a11, a, a22, …, a, …, ann) | a) | aii∈∈A for 1A for 1 ≤≤ ii ≤≤ n}n}
Lecture Slides By Adil Aslam

NoteNote
 We use the notation A² to denote A×A,We use the notation A² to denote A×A,
the Cartesian product of the set A withthe Cartesian product of the set A with
itself.itself.
 Similarly, A³=A×A×A, A =A×A×A×A, and so⁴Similarly, A³=A×A×A, A =A×A×A×A, and so⁴
on….on….
 More generallyMore generally
 A ={(aᴺA ={(aᴺ 11, a, a22, ...,an) | ai A for i =1,2,...,n}∈, ...,an) | ai A for i =1,2,...,n}∈
 ExampleExample
 Suppose that A={1,2}Suppose that A={1,2}
 A²= (1,1),(1,2),(2,2),(2,1)A²= (1,1),(1,2),(2,2),(2,1)
Lecture Slides By Adil Aslam

Overlapping SetOverlapping Set
• Two sets that have at least one
common element are called overlapping
sets.
• In case of overlapping sets :
n(A B) = n(A) + n(B) − n(A B)∪ ∩
n(A B) = n(A − B) + n(B − A) + n(A B)∪ ∩
n(A) = n(A − B) + n(A B)∩
n(B) = n(B − A) + n(A B)∩
Example: Let, A = {1, 2, 6} and B = {6, 12, 42}.
There is a common element ‘6’, hence these
sets are overlapping sets.
Lecture Slides By Adil Aslam

Disjoint SetDisjoint Set
If two sets C and D are disjoint sets as
they do not have even one element in
common. Therefore, n(A B) = n(A) +∪
n(B)
Example: Let, A = {1, 2, 6} and B = {7, 9,
14}, there is no common element, hence
these sets are overlapping sets.
Lecture Slides By Adil Aslam

ExerciseExercise
1.1. List the members of these sets.List the members of these sets.
a) {x | x is a real number such that x² = 1}a) {x | x is a real number such that x² = 1}
Ans:Ans: {-1, 1}{-1, 1}
b) {x | x is a positive integer less than 12}b) {x | x is a positive integer less than 12}
Ans:Ans: {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}
c) {x | x is the square of an integer and x < 100}c) {x | x is the square of an integer and x < 100}
Ans:Ans: {0, 1, 4, 9, 16, 25, 36, 49, 64, 81}{0, 1, 4, 9, 16, 25, 36, 49, 64, 81}
d) {x | x is an integer such that x² = 2}d) {x | x is an integer such that x² = 2}
Ans:Ans: ∅∅
Lecture Slides By Adil Aslam

ExerciseExercise
Use set builder notation to give a description ofUse set builder notation to give a description of
each of these sets.each of these sets.
a)a){0{0,, 33,, 66,, 99,, 12}12}
Ans:Ans:
b)b) {−3{−3,,−2−2,,−1−1,, 00,, 11,, 22,, 3}3}
Ans:Ans:
c)c) {{m, n, o, pm, n, o, p}}
Ans:Ans:
Lecture Slides By Adil Aslam

ExerciseExercise
For each of these pairs of sets, determineFor each of these pairs of sets, determine
whether the first is a subset of the second,whether the first is a subset of the second,
the second is a subset of the first, orthe second is a subset of the first, or
neither is a subset of the other.neither is a subset of the other.
a)a) the set of airline flights from New York tothe set of airline flights from New York to
New Delhi,New Delhi,
the set of nonstop airline flights from Newthe set of nonstop airline flights from New
York to New DelhiYork to New Delhi
Ans:Ans:
The first is a subset of the second, but theThe first is a subset of the second, but the
second is not a subset of the first.second is not a subset of the first.
Lecture Slides By Adil Aslam

ExerciseExercise
b)b) the set of people who speak English, the set ofthe set of people who speak English, the set of
people who speak Chinesepeople who speak Chinese
Ans:Ans:
Neither is a subset of the other.Neither is a subset of the other.
c)c) the set of flying squirrels, the set of livingthe set of flying squirrels, the set of living
creatures that can flycreatures that can fly
Ans:Ans:
The first is a subset of the second, but theThe first is a subset of the second, but the
second is not a subset of the first.second is not a subset of the first.
Lecture Slides By Adil Aslam

ExerciseExercise
a) the set of people who speak English, the set ofa) the set of people who speak English, the set of
people who speak English with an Australian accentpeople who speak English with an Australian accent
Ans:Ans:
{{ people who speak Englishpeople who speak English } {⊇} {⊇ people who speakpeople who speak
English with an Australian accentEnglish with an Australian accent }}
b)b) the set of fruits, the set of citrus fruitsthe set of fruits, the set of citrus fruits
Ans:Ans:
{{ fruitsfruits } {⊇} {⊇ citrus fruitscitrus fruits }}
c)c) the set of students studying discretethe set of students studying discrete
mathematics, the set of students studying datamathematics, the set of students studying data
structuresstructures
Ans:Ans:
Normally, neither is a subset of the other.Normally, neither is a subset of the other.
Lecture Slides By Adil Aslam

ExerciseExercise
Determine whether each of these pairs of setsDetermine whether each of these pairs of sets
are equal.are equal.
a) {1a) {1,, 33,, 33,, 33,, 55,, 55,, 55,, 55,, 5}5},,{5{5,, 33,, 1}1}
Ans:Ans:
YesYes
b)b) {{1}}{{1}},,{1{1,,{1}}{1}}
Ans:Ans:
NoNo
c)c) ∅∅,,{ }∅{ }∅
Ans:Ans:
NoNo
Lecture Slides By Adil Aslam

ExerciseExercise
Suppose that A = {2, 4, 6}, B = {2, 6}, C = {4, 6},
and D = {4, 6, 8}. Determine which of these sets
are subsets of which other of these sets.
Ans:
Every set is a subset of itselfEvery set is a subset of itself
A and D are not subsets of any other set listed.A and D are not subsets of any other set listed.
B is a subset of A.B is a subset of A.
C is a subset of both A and D.C is a subset of both A and D.
Lecture Slides By Adil Aslam

ExerciseExercise
For each of the following sets, determineFor each of the following sets, determine
whether 2 is an element of that set.whether 2 is an element of that set.
a) { x R|x is an integer greater than 1}∈a) { x R|x is an integer greater than 1}∈
Ans :Ans :
YesYes
b) { x R|x is the square of an integer}∈b) { x R|x is the square of an integer}∈
Ans :Ans :
NoNo
Lecture Slides By Adil Aslam

ExerciseExercise
c) {2,{2}}c) {2,{2}}
Ans :Ans :
YesYes
d) {{2},{{2}}}d) {{2},{{2}}}
Ans :Ans :
NoNo
e) {{2},{2,{2}}}e) {{2},{2,{2}}}
Ans :Ans :
NoNo
Lecture Slides By Adil Aslam

ExerciseExercisef) {{{2}}f) {{{2}}
Ans :Ans :
NoNo
Determine whether each of these statements isDetermine whether each of these statements is
true or false.true or false.
a) 0 ∈ ∅a) 0 ∈ ∅
Ans :Ans :
False (The empty set has no elements)False (The empty set has no elements)
b) {0}∅ ∈b) {0}∅ ∈
Ans :Ans :
False (The empty set is a subset of {0}, but isFalse (The empty set is a subset of {0}, but is
not an element of it).not an element of it).Lecture Slides By Adil Aslam

ExerciseExercise
c) {0} ⊂ ∅c) {0} ⊂ ∅
Ans :Ans :
False (No set can be a proper subset ofFalse (No set can be a proper subset of
the empty set since, by definition, that wouldthe empty set since, by definition, that would
require the empty set to contain at least onerequire the empty set to contain at least one
element).element).
d) {0}∅ ⊂d) {0}∅ ⊂
Ans :Ans :
True (The empty set is a subset ofTrue (The empty set is a subset of
every set, {0} . In addition the set {0} has∅ ⊂every set, {0} . In addition the set {0} has∅ ⊂
one element, which is not contained in the emptyone element, which is not contained in the empty
set).set). Lecture Slides By Adil Aslam

ExerciseExercisee) {0} {0}∈e) {0} {0}∈
Ans :Ans :
False (The set {0} is a subset of itself,False (The set {0} is a subset of itself,
but is not an element of itself).but is not an element of itself).
f) {0} {0}⊂f) {0} {0}⊂
Ans:Ans:
False (0} Í {0} since {0} = {0}. But, asFalse (0} Í {0} since {0} = {0}. But, as
noted above, the proper subset relation requiresnoted above, the proper subset relation requires
the larger set to have at least one element notthe larger set to have at least one element not
in the other one).in the other one).
g) { } { }∅ ⊆ ∅g) { } { }∅ ⊆ ∅
Ans :Ans :
True (Every set is a subset of itself)True (Every set is a subset of itself)
Lecture Slides By Adil Aslam

ExerciseExercise
 Determine whether these statementsDetermine whether these statements
are true or falseare true or false..
a) { }∅ ∈ ∅a) { }∅ ∈ ∅
Ans :Ans :
True ( the set { } has exactly one element,∅True ( the set { } has exactly one element,∅
namely ).∅namely ).∅
b) { ,{ }∅ ∈ ∅ ∅b) { ,{ }∅ ∈ ∅ ∅
Ans :Ans :
True (The empty set is one of the twoTrue (The empty set is one of the two
elements of that set).elements of that set).
Lecture Slides By Adil Aslam

ExerciseExercisec) { } { }∅ ∈ ∅c) { } { }∅ ∈ ∅
Ans :Ans :
False (A set is a subset of itself, but is not anFalse (A set is a subset of itself, but is not an
element of itself).element of itself).
d) { } {{ }}∅ ∈ ∅d) { } {{ }}∅ ∈ ∅
AnsAns
True (Here the set which has the empty set asTrue (Here the set which has the empty set as
its only element is, in turn, the only element of the setits only element is, in turn, the only element of the set
on the right).on the right).
e) { } { ,{ }}∅ ⊂ ∅ ∅e) { } { ,{ }}∅ ⊂ ∅ ∅
Ans :Ans :
True (The set on the left has the empty set asTrue (The set on the left has the empty set as
its only element and the empty set occurs as an elementits only element and the empty set occurs as an element
of the set on the right, which also contains a secondof the set on the right, which also contains a second
element, namely { } .∅element, namely { } .∅ Lecture Slides By Adil Aslam

ExerciseExercise
f) {{ }} { ,{ }}∅ ⊂ ∅ ∅f) {{ }} { ,{ }}∅ ⊂ ∅ ∅
Ans :Ans :
True ( the set on the left hasTrue ( the set on the left has
{ } as its only element, which occurs as∅{ } as its only element, which occurs as∅
one of the two elements of the set on theone of the two elements of the set on the
right).right).
g) {{ }} {{ },{ }}∅ ⊂ ∅ ∅g) {{ }} {{ },{ }}∅ ⊂ ∅ ∅
Ans :Ans :
FalseFalse
Lecture Slides By Adil Aslam

ExerciseExercise
Determine whether each of these statements isDetermine whether each of these statements is
true or false.true or false.
a) x {x}∈a) x {x}∈
Ans :Ans :
True (The set {x} has exactly one element,True (The set {x} has exactly one element,
namely x).namely x).
b){x} {x}.⊆b){x} {x}.⊆
Ans :Ans :
True (every set is a subset of itself).True (every set is a subset of itself).
c) {x} {x}∈c) {x} {x}∈
Ans :Ans :
False (A set is a subset of itself, but isFalse (A set is a subset of itself, but is
not an element of itself).not an element of itself). Lecture Slides By Adil Aslam

ExerciseExercise
{x} {{x}}∈{x} {{x}}∈
Ans :Ans :
True ( Here the set which has x as its onlyTrue ( Here the set which has x as its only
element is, in turn, the only element of the setelement is, in turn, the only element of the set
on the right).on the right).
e) {x}∅⊆e) {x}∅⊆
Ans :Ans :
True (The empty set is a subset of every set).True (The empty set is a subset of every set).
f) {x}∅ ∈f) {x}∅ ∈
Ans :Ans :
False ( The only element of the set {x} is x).False ( The only element of the set {x} is x).
Lecture Slides By Adil Aslam

ExerciseExercise
Use a Venn diagram to illustrate the subset ofUse a Venn diagram to illustrate the subset of
odd integers in the set of all positive integers notodd integers in the set of all positive integers not
exceeding 10.exceeding 10.
Ans:Ans:
Lecture Slides By Adil Aslam

ExerciseExercise
Use a Venn diagram to illustrate the set of allUse a Venn diagram to illustrate the set of all
months of the year whose names do not containmonths of the year whose names do not contain
the letter R in the set of all months of thethe letter R in the set of all months of the
year.year.
Ans :Ans :
Lecture Slides By Adil Aslam

ExerciseExercise
Use a Venn diagram to illustrate the relationshipUse a Venn diagram to illustrate the relationship
AA ⊆⊆ BB andand BB ⊆⊆ C.C.
Ans:Ans:
Lecture Slides By Adil Aslam

ExerciseExercise
Use a Venn diagram to illustrate the relationshipsUse a Venn diagram to illustrate the relationships
AA ⊂⊂ BB andand BB ⊂⊂ CC..
Ans:Ans:
Lecture Slides By Adil Aslam

ExerciseExercise
Find two sets A and B such that A B and∈Find two sets A and B such that A B and∈
A B⊆A B⊆
Ans:Ans:
A = Ø and B = {Ø}, such that Ø {Ø} and∈A = Ø and B = {Ø}, such that Ø {Ø} and∈
Ø {Ø}, As we previously know⊆Ø {Ø}, As we previously know⊆
that Ø is a subset of any set.that Ø is a subset of any set.
Lecture Slides By Adil Aslam

ExerciseExercise
 What is the cardinality of each of theseWhat is the cardinality of each of these
sets?sets?
a) {a}a) {a}
Ans :Ans :
|a| =|a| = 11
b) {{a}}b) {{a}}
Ans :Ans :
|b|= 1|b|= 1
c) {a,{a}}c) {a,{a}}
Ans :Ans :
|c|= 2|c|= 2
Lecture Slides By Adil Aslam

ExerciseExercise
d){a,{a},{a,{a}}}d){a,{a},{a,{a}}}
Ans :Ans :
|d|= 3|d|= 3
 What is the cardinality of each of theseWhat is the cardinality of each of these
sets?sets?
a) ∅a) ∅
Ans :Ans :
|a|= 0|a|= 0
b){ }∅b){ }∅
Ans :Ans :
|b| = 1|b| = 1
Lecture Slides By Adil Aslam

ExerciseExercise
c) { ,{ }}∅ ∅c) { ,{ }}∅ ∅
Ans :Ans :
|c| = 2|c| = 2
d) { ,{ },{ ,{ }}}∅ ∅ ∅ ∅d) { ,{ },{ ,{ }}}∅ ∅ ∅ ∅
Ans :Ans :
|d| = 3|d| = 3
 Find the power set of each of these sets, where aFind the power set of each of these sets, where a
and b are distinct elements.and b are distinct elements.
a) {a}a) {a}
Ans :Ans :
P(a) = { , {a}}∅P(a) = { , {a}}∅
Lecture Slides By Adil Aslam

ExerciseExercise
b) {a, b}b) {a, b}
Ans :Ans :
P(b) = { , {a}, {b}, {a, b}}∅P(b) = { , {a}, {b}, {a, b}}∅
c) { ,{ }}∅ ∅c) { ,{ }}∅ ∅
Ans :Ans :
P(c) = { ,{ },{{ }},{ ,{ }}}∅ ∅ ∅ ∅ ∅P(c) = { ,{ },{{ }},{ ,{ }}}∅ ∅ ∅ ∅ ∅
Lecture Slides By Adil Aslam

ExerciseExercise
Can you conclude that A=B if A and B are twoCan you conclude that A=B if A and B are two
sets with the same power set?sets with the same power set?
Ans :Ans :
Yes. By definition, P(A) is the set of allYes. By definition, P(A) is the set of all
subsets that can be generated from A, if A and Bsubsets that can be generated from A, if A and B
generate the exact same collection of validgenerate the exact same collection of valid
subsets, then it must be that A and B contain thesubsets, then it must be that A and B contain the
same elements and are therefore equal.same elements and are therefore equal.
Lecture Slides By Adil Aslam

ExerciseExercise
How many elements does each of these sets haveHow many elements does each of these sets have
WhereWhere aa andand bb are distinct elements?are distinct elements?
a) P(a) P({{a, b,a, b, {{a, ba, b}}}}))
Ans:Ans:
|{a, b, {a, b}}||{a, b, {a, b}}| = 3, so= 3, so |P|P(({a, b, {a, b}}{a, b, {a, b}}))|| = 2= 2³³ = 8.= 8.
b)b) P(P({∅{∅, a,, a, {{aa}},, {{{{aa}}}}}}))
Ans:Ans:
|{ , a, {a}, {{a}}}|∅|{ , a, {a}, {{a}}}|∅ = 4, so= 4, so |P|P(({ , a, {a}, {{a}}}∅{ , a, {a}, {{a}}}∅ ))|| = 2⁴= 2⁴
= 16.= 16.
c)c) P(P(P(P(∅∅))))
Ans:Ans:
|P|P((∅∅))|| = 1, so= 1, so |P|P((∅∅))|| = 2.= 2.Lecture Slides By Adil Aslam

ExerciseExercise
Determine whether each of these sets is theDetermine whether each of these sets is the
power set of a set, wherepower set of a set, where aa andand bb are distinctare distinct
elements.elements.
a)a) ∅∅
Ans:Ans:
No setNo set
b)b) {∅{∅,, {{aa}}}}
Ans:Ans:
This is a power set of the set { a }This is a power set of the set { a }
Lecture Slides By Adil Aslam

ExerciseExercise
c)c) {∅{∅,, {{aa}},, {∅{∅, a, a}}}}
Ans:Ans:
This is not a power set, since it has 3 elementsThis is not a power set, since it has 3 elements
and the number of elements in a power set is 2^n,and the number of elements in a power set is 2^n,
where n is the number of elements in the set.where n is the number of elements in the set.
d)d) {∅{∅,, {{aa}},, {{bb}},, {{a, ba, b}}}}
Ans:Ans:
Power set of {a, b}Power set of {a, b}
Lecture Slides By Adil Aslam

ExerciseExercise
LetLet AA = {= {a, b, c, da, b, c, d} and} and BB = {= {y, zy, z}. Find}. Find
a) Aa) A ×× BB
Ans:Ans:
There will be 4 x 2 = 8 sets.There will be 4 x 2 = 8 sets.
{(a,y), (a,z), (b,y), (b,z), (c,y), (c,z), (d,y), (d,z)}{(a,y), (a,z), (b,y), (b,z), (c,y), (c,z), (d,y), (d,z)}
b)b) BB ×× AA
Ans:Ans:
{(y,a), (y,b), (y,c), (y,d), (z,a), (z,b), (z,c), (z,d)}{(y,a), (y,b), (y,c), (y,d), (z,a), (z,b), (z,c), (z,d)}
Lecture Slides By Adil Aslam

ExerciseExercise
What is the Cartesian product A×B, where Ai sWhat is the Cartesian product A×B, where Ai s
the set of courses offered by the mathematicsthe set of courses offered by the mathematics
department at a university and B is the set ofdepartment at a university and B is the set of
mathematics professors at this university? Givemathematics professors at this university? Give
an example of how this Cartesian product can bean example of how this Cartesian product can be
usedused..
Ans:Ans:
It is the set of all possible combinations of mathIt is the set of all possible combinations of math
courses and possible instructors.courses and possible instructors.
Lecture Slides By Adil Aslam

ExerciseExercise
 What is the Cartesian product A×B×C, whereWhat is the Cartesian product A×B×C, where
A is the set of all airlines and B and CareA is the set of all airlines and B and Care
both the set of all cities in the Unitedboth the set of all cities in the United
States? Give an example of how thisStates? Give an example of how this
Cartesian product can be used.Cartesian product can be used.
Ans :Ans :
The set of triples (a,b,c), where a is anThe set of triples (a,b,c), where a is an
airline and b and c are cities. A usefulairline and b and c are cities. A useful
subset of this set is the set ofsubset of this set is the set of
triples(a,b,c) for which a flies between btriples(a,b,c) for which a flies between b
and c.and c.
Lecture Slides By Adil Aslam

ExerciseExercise
Suppose thatSuppose that AA ×× BB = , where∅= , where∅ AA andand BB are sets.are sets.
What can you conclude?What can you conclude?
Ans:Ans:
If bothIf both AA andand BB are nonempty, then this isn’tare nonempty, then this isn’t
possible. So at least one ofpossible. So at least one of AA
oror BB must be empty. And in fact, ifmust be empty. And in fact, if AA is empty,is empty,
then so isthen so is A × BA × B; similarly so when; similarly so when BB
is empty.is empty.
Lecture Slides By Adil Aslam

ExerciseExercise
LetLet AA be a set. Show that ×∅be a set. Show that ×∅ AA == AA × = .∅ ∅× = .∅ ∅
Ans:Ans:
∅∅ x A = {(x, y) | x and y A} = = {(x, y) | x∈ ∅ ∈ ∅x A = {(x, y) | x and y A} = = {(x, y) | x∈ ∅ ∈ ∅
∈∈ A and y } = A x∈ ∅ ∅A and y } = A x∈ ∅ ∅
OROR
∅∅×A={(x, y)| x and∈ ∅×A={(x, y)| x and∈ ∅
y A} = ={(x, y)| x A and∈ ∅ ∈y A} = ={(x, y)| x A and∈ ∅ ∈
y }=A×∈∅ ∅y }=A×∈∅ ∅
Lecture Slides By Adil Aslam

ExerciseExercise
Let A={a, b, c}, B={x,y}, and C={0,1}. FindLet A={a, b, c}, B={x,y}, and C={0,1}. Find
a) A×B×Ca) A×B×C
Ans :Ans :
A×B×C = (A×B×C = (a, x, 0) ,(a, y, 0) ,(a, x, 1) ,(a, y, 1) ,(b,a, x, 0) ,(a, y, 0) ,(a, x, 1) ,(a, y, 1) ,(b,
x, 0) ,(b, y, 0) (b, x, 1),(b ,y, 1), (c, x, 0), (c, y, 0),x, 0) ,(b, y, 0) (b, x, 1),(b ,y, 1), (c, x, 0), (c, y, 0),
(c, x, 1) ,(c, y, 1)(c, x, 1) ,(c, y, 1)
b)C×B×Ab)C×B×A
Ans :Ans :
C×B×A ={(0,x,a),C×B×A ={(0,x,a),(0,x,b),(0,x,c),(1,x,a),(1,x,b),(0,x,b),(0,x,c),(1,x,a),(1,x,b),
(1,x,c),(0,y,a),(1,x,c),(0,y,a),
(0,y,b),(0,y,c),(1,y,a),(1,y,b),(1,y,c)}.(0,y,b),(0,y,c),(1,y,a),(1,y,b),(1,y,c)}.
Lecture Slides By Adil Aslam

ExerciseExercise
d) B×B×Bd) B×B×B
Ans :Ans : B×B×B=B×B×B={(x,x,x),(x,x,y),(x,y,y),(x,y,x),{(x,x,x),(x,x,y),(x,y,y),(x,y,x),
(y,y,y),(y,y,y),
(y,x,x),(y,y,x),(y,x,y)}(y,x,x),(y,y,x),(y,x,y)}
Find A² ifFind A² if
a) A={0,1,3}a) A={0,1,3}
Ans :Ans : A²={(0,0),(0,1),(0,3),(1,0),(1,1),A²={(0,0),(0,1),(0,3),(1,0),(1,1),
(1,3),(3,0),(3,1),(3,3)}(1,3),(3,0),(3,1),(3,3)}
Lecture Slides By Adil Aslam

ExerciseExercise
b)A={1,2,a,b}b)A={1,2,a,b}
Ans :Ans :
{(1,1), (1,2), (1,a), (1,b), (2,1), (2,2), (2,a), (2,b), (a,1),{(1,1), (1,2), (1,a), (1,b), (2,1), (2,2), (2,a), (2,b), (a,1),
(a,2),(a, a),(a, b),(b,1),(b,2),(b, a),(b, b)}(a,2),(a, a),(a, b),(b,1),(b,2),(b, a),(b, b)}
Find A³ ifFind A³ if
a) A={a}a) A={a}
Ans :Ans :
A³= {(a, a, a)}A³= {(a, a, a)}
b) A={0,a}b) A={0,a}
Ans :Ans :
A³= {(0, 0, 0), (0, 0,1), (0,1, 0), (1, 0, 0), (1, 1,0),(1,0, 1),A³= {(0, 0, 0), (0, 0,1), (0,1, 0), (1, 0, 0), (1, 1,0),(1,0, 1),
(0, 1, 1), (1, 1, 1)}(0, 1, 1), (1, 1, 1)}
Lecture Slides By Adil Aslam

ExerciseExercise
How many different elements does A×B haveHow many different elements does A×B have
if A has m elements and B has n elements?if A has m elements and B has n elements?
Ans :Ans :
mnmn
How many different elements does A×B×CHow many different elements does A×B×C
have if A Has m elements has n elements, andhave if A Has m elements has n elements, and
C has p elements?C has p elements?
Ans :Ans :
mnpmnp
Lecture Slides By Adil Aslam

ExerciseExercise
How many different elements does A^n have when AHow many different elements does A^n have when A
has m elements and n is a positive integehas m elements and n is a positive integer?r?
Ans :Ans :
m^nm^n
Show that A×B≠ B×A, when A and B are nonempty,Show that A×B≠ B×A, when A and B are nonempty,
unless A=B.unless A=B.
Ans :Ans :
Lets us suppose that A= {a,b} B= {1,2}Lets us suppose that A= {a,b} B= {1,2}
A×B={(a,1),(a,2),(b,1),(b,2)}A×B={(a,1),(a,2),(b,1),(b,2)}
B×A={(1,a),(1,b),(2,a),(2,b)}B×A={(1,a),(1,b),(2,a),(2,b)}
Hence proved .Hence proved .
Lecture Slides By Adil Aslam

ExerciseExercise
Explain why A×B×C and (A×B)×C are notExplain why A×B×C and (A×B)×C are not
the same.the same.
Ans :Ans :
The elements of A×B×C consist of 3-The elements of A×B×C consist of 3-
tuples (a,b,c), where a A, b B, and c C,∈ ∈ ∈tuples (a,b,c), where a A, b B, and c C,∈ ∈ ∈
whereas the elements of (A×B)×C look likewhereas the elements of (A×B)×C look like
((a, b), c)—ordered pairs, the first((a, b), c)—ordered pairs, the first
coordinate of which is again an ordered pair.coordinate of which is again an ordered pair.
Lecture Slides By Adil Aslam

Any QuestionAny Question
??
Lecture Slides By Adil Aslam

Thank YouThank You 
Lecture Slides By Adil Aslam