SET

Sets, Relations
and Binary
Operations

Set
Set is a collection of well defined objects which are distinct from each
other. Sets are usually denoted by capital letters A, B,C,K and
elements are usually denoted by small letters a, b, c,... .
If a is an element of a set A, then we write a Î A and say a belongs to A
or a is in A or a is a member of A. If a does not belongs to A, we write
a ÏA.
Standard Notations
N : A set of natural numbers.
W : A set of whole numbers.
Z : A set of integers.
Z+ / Z− : A set of all positive/negative integers.
Q : A set of all rational numbers.
Q+ / Q− : A set of all positive/negative rational numbers.
R : A set of real numbers.
R+ / R− : A set of all positive/negative real numbers.
C : A set of all complex numbers.

Methods for Describing a Set

(i) Roster/Listing Method/Tabular Form In this method, a
set is described by listing element, separated by commas, within
braces.
e.g. A = {a, e, i, o, u}
(ii) Set Builder/Rule Method In this method, we write down a
property or rule which gives us all the elements of the set by that
rule.
e.g. A = { x : x is a vowel of English alphabets}

Types of Sets

(i) Finite Set A set containing finite number of elements or no
element.
(ii) Cardinal Number of a Finite Set The number of elements
in a given finite set is called cardinal number of finite set,
denoted by n (A).
(iii) Infinite Set A set containing infinite number of elements.
(iv) Empty/Null/Void Set A set containing no element, it is
denoted by f or { }.
(v) Singleton Set A set containing a single element.
(vi) Equal Sets Two sets A and B are said to be equal, if every
element of A is a member of B and every element of B is a member
of A and we write A= B.
(vii) Equivalent Sets Two sets are said to be equivalent, if they
have same number of elements.
If n(A) = n(B), then A and B are equivalent sets.
(viii) Subset and Superset Let A and B be two sets. If every
element of A is an element of B, then A is called subset of B and B
is called superset of A.
Written as A Í B or B Ê A
(ix) Proper Subset If A is a subset of B and A ¹ B, then A is called
proper subset of B and we write A Ì B.
(x) Universal Set (U) A set consisting of all possible elements
which occurs under consideration is called a universal set.
(xi) Comparable Sets Two sets A and Bare comparable, if A Í B
or B Í A.
(xii) Non-Comparable Sets For two sets A and B, if neither
A Í B nor B Í A, then A and B are called non-comparable sets.
(xiii) Power Set The set formed by all the subsets of a given set A, is
called power set of A, denoted by P(A).
(xiv) Disjoint Sets Two sets A and B are called disjoint, if
AÇ B = f. i.e. they do not have any common element.

Venn Diagram

In a Venn diagram, the universal set is

represented by a rectangular region and a

set is represented by circle or a closed

geometrical figure inside the universal set.

Operations on Sets

1. Union of Sets

The union of two sets A and B, denoted by
A È B, is the set of all those elements, each
one of which is either in A or in B or both in
A and B.

     2. Intersection of Sets
The intersection of two sets A and B,
denoted by A Ç B, is the set of all those
elements which are common to both
A and B.
If A A A1 2 n , ,... , is a finite family of sets,
then their intersection is denoted by
Ç Ç Ç Ç
i =n i n A A A A 1 1 2 or ...

3. Complement of a Set

If A is a set with U as universal set, then
complement of a set A, denoted by A¢ or Ac is the
setU − A.

4. Difference of Sets
For two sets A and B, the difference A − B is the
set of all those elements of A which do not belong
to B.

5. Symmetric Difference 

For two sets A and B, symmetric difference is the set (A − B) È (B − A)denoted by A D B.

Laws of Algebra of Sets
For three sets A, B and C
(i) Idempotent Law

(a) A È A = A                           (b) A Ç A = A

(ii) Identity Law

(a) A È f = A                            (b) A ÇU = A

(iii) Commutative Law

(a) A È B = B È A                   (b) A Ç B = B Ç A

(iv) Associative Law

(a) (A È B) È C = A È (B È C)
(b) A Ç (B Ç C) = (A Ç B) Ç C

(v) Distributive Law

(a) A È (B Ç C) = (A È B) Ç (A È C)
(b) A Ç (B È C) = (A Ç B) È (A Ç C)

(vi) De-Morgan’s Law

(a) (A È B)¢ = A¢ Ç B¢
(b) (A Ç B)¢ = A¢ È B¢
(vii) (a) A − (B Ç C) = (A − B) È (A − C)
(b) A − (B È C) = (A − B) Ç ( A − C)

(viii) (a) A − B = A Ç B¢ (b) B − A = B Ç A¢

(c) A − B = AÛ A Ç B = f (d) (A − B) È B = A È B
(e) (A − B) Ç B = f (f) A Ç B Í A and A Ç B Í B
(g) AÈ(AÇ B) = A (h) AÇ(AÈ B) = A

(ix) (a) (A − B) È (B − A) = (A È B) − (A Ç B)
(b) A Ç (B − C) = (A Ç B) − (A Ç C)
(c) A Ç (B DC) = (A Ç B) D (A Ç C)
(d) (AÇ B)È(A − B) = A
(e) AÈ(B − A) = (AÈ B)

(x) (a)U ¢ = f
(b) f¢ =U
(c) (A¢ )¢ = A
(d) A Ç A¢ = f
(e) A È A¢ =U
(f) A Í BÛ B¢ Í A¢

Important Points to be Remembered
(i) Every set is a subset of itself i.e. A Í A, for any set A.
(ii) Empty set f is a subset of every set i.e. f Ì A, for any set A.
(iii) For any set A and its universal setU, A ÍU
(iv) If A = f, then power set has only one element
i.e. n(P(A)) =1
(v) Power set of any set is always a non-empty set.
(vi) Suppose A = {1, 2}, then
P(A) = {{1}, {2}, {1, 2}, f}
(a) A ÏP(A)
(b) {A} ÎP(A)
(vii) If a set A hasn elements, then P(A) or subset of A has 2n elements.
(viii) Equal sets are always equivalent but equivalent sets may not be equal.
(ix) The set {f} is not a null set. It is a set containing one element f.

Results on Number of Elements in Sets
(i) n (A È B) = n(A) + n (B) − n(A Ç B)
(ii) n(A È B) = n(A) + n(B), if A and B are disjoint sets.
(iii) n(A − B) = n(A) − n(A Ç B)
(iv) n (B− A) = n (B) − n (AÇ B)
(v) n(A D B) = n(A) + n(B) − 2n(A Ç B)
(vi) n(A È B È C) = n(A) + n(B) + n(C) − n(A Ç B)
− n(B Ç C) − n(A Ç C) + n(A Ç B Ç C)
(vii) n (number of elements in exactly two of the sets A, B,C)
= n(A Ç B) + n(B Ç C) + n(C Ç A) − 3n(A Ç B Ç C)
(viii) n (number of elements in exactly one of the sets A, B,C)
= n(A) + n(B) + n(C) − 2n(A Ç B)
− 2n(B Ç C) − 2n(A Ç C) + 3n(A Ç B Ç C)
(ix) n(A¢ È B¢ ) = n(A Ç B)¢ = n(U ) − n(A Ç B)
(x) n(A¢ Ç B¢ ) = n(A È B)¢ = n(U ) − n(A È B)

Ordered Pair
An ordered pair consists of two objects or elements in a given fixed
order.
Equality of Ordered Pairs Two ordered pairs (a , b ) 1 1 and (a , b ) 2 2
are equal, iff a a 1 2 = and b b 1 2 = 

Cartesian Product of Sets
For two sets A and B (non-empty sets), the set of all ordered pairs (a, b)
such that a Î A and bÎB is called Cartesian product of the sets A and
B, denoted by A × B.
A × B = {(a, b) : a Î A and bÎB}
If there are three sets A, B, C and aÎA, bÎB and cÎC, then we form
an ordered triplet (a, b, c). The set of all ordered triplets (a, b, c) is
called the cartesian product of these sets A, B and C.
i.e. A× B×C = {(a,b, c):aÎA,bÎB, cÎC}

Properties of Cartesian Product
For three sets A, B and C
(i) n (A × B) = n(A) n(B)
(ii) A × B = f, if either A or B is an empty set.
(iii) A × (B È C) = (A × B) È (A × C)
(iv) A × (B Ç C) = (A × B) Ç (A × C)
(v) A × (B − C) = (A × B) − (A × C)
(vi) (A × B) Ç (C × D) = (A Ç C) × (B Ç D)
(vii) If A Í B and C Í D, then (A × C) Í (B × D)
(viii) If A Í B, then A × A Í (A × B) Ç (B × A)
(ix) A × B = B × AÛ A = B
(x) If either A or B is an infinite set, then A × B is an infinite set.
(xi) A × (B¢ È C¢ )¢ = (A × B) Ç (A × C)
(xii) A × (B¢ Ç C¢ )¢ = (A × B) È (A × C)
(xiii) If A and B be any two non-empty sets having n elements in
common, then A × Band B × Ahave n2 elements in common.
(xiv) If A¹ B, then A × B¹ B × A
(xv) If A = B, then A × B= B × A
(xvi) If A Í B, then A ×C Í B ×C for any set C.

Relation
If A and B are two non-empty sets, then a relation R from A to B is a
subset of A × B .
If R Í A × B and (a, b)ÎR, then we say that a is related to b by the
relation R, written as aRb.
Domain and Range of a Relation
Let R be a relation from a set A to set B. Then, set of all first
components or coordinates of the ordered pairs belonging to R is called
the domain of R, while the set of all second components or coordinates
of the ordered pairs belonging to R is called the range of R.
Thus, domain of R = {a : (a, b)ÎR} and range of R = { b : (a, b)ÎR}

Types of Relation
(i) Void Relation As f Ì A × A, for any set A, so f is a relation on
A, called the empty or void relation.
(ii) Universal Relation Since, A × A Í A × A, so A × A is a
relation on A, called the universal relation.
(iii) Identity Relation The relation I a a a A A = {( , ) : Î } is called
the identity relation on A.
(iv) Reflexive Relation A relation R is said to be reflexive
relation, if every element of A is related to itself.
Thus, (a, a)ÎR," a Î AÞ R is reflexive.
(v) Symmetric Relation A relation R is said to be symmetric
relation, iff
(a, b)ÎR Þ(b, a)ÎR," a, bÎ A
i.e. a R bÞ bRa," a, bÎ A
ÞR is symmetric.
(vi) Anti-Symmetric Relation A relation R is said to be
anti-symmetric relation, iff
(a, b)ÎR and (b, a)ÎR Þa = b," a, bÎ A
(vii) Transitive Relation A relation R is said to be transitive
relation, iff (a, b)ÎR and (b, c)ÎR
Þ (a, c)ÎR," a, b, c Î A
(viii) Equivalence Relation A relation R is said to be an
equivalence relation, if it is simultaneously reflexive, symmetric
and transitive on A.
(ix) Partial Order Relation A relation R is said to be a partial
order relation, if it is simultaneously reflexive, symmetric and
anti-symmetric on A.
(x) Total Order Relation A relation R on a set A is said to be a
total order relation on A, if R is a partial order relation on A.

Inverse Relation
If A and B are two non-empty sets and R be a relation from A to B,
such that R = {(a, b) : a Î A, bÎB}, then the inverse of R, denoted by
R−1, is a relation from B to A and is defined by
R−1 = b a a b ÎR {( , ) : ( , ) }
Equivalence Classes of an Equivalence Relation
Let R be equivalence relation in A (¹ f). Let aÎA .
Then, the equivalence class of a denoted by [a] or {a} is defined as the
set of all those points of A which are related to a under the relation R.

Composition of Relation
Let R and S be two relations from sets A to B and B to C respectively,
then we can define relation SoR from A to C such that
(a, c)ÎSoR Û $ bÎB such that (a, b)ÎR and (b, c)ÎS.
This relation SoR is called the composition of R and S.
(i) RoS ¹ SoR
(ii) (SoR)−1 = R−1oS−1 known as reversal rule.
Congruence Modulo m
Let m be an arbitrary but fixed integer. Two integers a and b are said
to be congruence modulo m, if a − b is divisible by m and we write a º b
(mod m).
i.e. a º b (mod m) Û a − b is divisible by m.
Important Results on Relation
(i) If R and S are two equivalence relations on a set A, then R Ç S is also on
equivalence relation on A.
(ii) The union of two equivalence relations on a set is not necessarily an
equivalence relation on the set.
(iii) If R is an equivalence relation on a set A, then R−1 is also an equivalence
relation on A.
(iv) If a set A hasn elements, then number of reflexive relations from A to A is
2
2 n − n.
(v) Let AandB be two non-empty finite sets consisting ofmand n elements,
respectively. Then, A×B consists of mn ordered pairs. So, the total
number of relations from A to B is 2nm.

Binary Operations
Let S be a non-empty set. A function f fromS ×S to S is called a binary
operation on S i.e. f : S ×S®S is a binary operation on set S.
Closure Property
An operation * on a non-empty set S is said to satisfy the closure
property, if
aÎS,bÎS Þa* bÎS, " a, bÎS
Also, in this case we say that S is closed for *.
An operation * on a non-empty set S, satisfying the closure property is
known as a binary operation.
Properties
(i) Generally binary operations are represented by the symbols * ,
Å, ... etc.,  instead of letters figure etc.
(ii) Addition is a binary operation on each one of the sets N, Z, Q, R
and C of natural numbers, integers, rationals, real and complex
numbers, respectively. While addition on the set S of all
irrationals is not a binary operation.
(iii) Multiplication is a binary operation on each one of the sets N, Z,
Q, R and C of natural numbers, integers, rationals, real and
complex numbers, respectively. While multiplication on the set S
of all irrationals is not a binary operation.
(iv) Subtraction is a binary operation on each one of the sets Z, Q, R
and C of integers, rationals, real and complex numbers,
respectively. While subtraction on the set of natural numbers is
not a binary operation.
(v) Let S be a non-empty set and P S ( ) be its power set. Then, the
union, intersection and difference of sets, on P S ( ) is a binary
operation.
(vi) Division is not a binary operation on any of the sets N, Z, Q, R
and C. However, it is not a binary operation on the sets of all
non-zero rational (real or complex) numbers.
(vii) Exponential operation (a, b)® ab is a binary operation on setNof
natural numbers while it is not a binary operation on set Z of
integers.

Types of Binary Operations
(i) Associative Law  A binary operation * on a non-empty set S is
said to be associative, if (a * b) * c = a * (b * c), " a, b, c ÎS.
Let R be the set of real numbers, then addition and
multiplication on R satisfies the associative law.
(ii) Commutative Law A binary operation * on a non-empty set S
is said to be commutative, if
a * b = b * a, " a, b ÎS.
Addition and multiplication are commutative binary operations
on Z but subtraction not a commutative binary operation, since
2 − 3 ¹ 3 − 2 .
Union and intersection are commutative binary operations on
the power set P S ( ) of all subsets of set S. But difference of sets is
not a commutative binary operation on P S ( ).
(iii) Distributive Law Let * and o be two binary operations on a
non-empty sets. We say that * is distributed over o., if
a* (b o c) = (a* b) o (a* c)," a, b, cÎS also called (left distribution)
and (b o c) * a = (b * a) o (c * a), " a, b, c Î S also called (right
distribution).
Let R be the set of all real numbers, then multiplication
distributes addition on R.
Since, a × (b + c) = a × b + a × c, " a, b, c ÎR.
(iv) Identity Element Let * be a binary operation on a non-empty
set S. An element e ÎS, if it exist such that
a*e = e*a = a, " a Î S.
is called an identity elements of S, with respect to *.
For addition on R, zero is the identity elements in R.
Since, a + 0 = 0 + a = a, " aÎR
For multiplication on R, 1 is the identity element in R.
Since, a × 1 = 1 × a = a," aÎR
Let P (S) be the power set of a non-empty set S. Then, f is the
identity element for union on P (S) as
AÈ f = fÈ A= A," AÎP(S)

Also, S is the identity element for intersection on P S ( ).
Since, AÇS = AÇS = A, " AÎP (S).
For addition on N the identity element does not exist. But for
multiplication on N the identity element is 1.
(v) Inverse of an Element Let * be a binary operation on a
non-empty set S and let e be the identity element.
Let aÎS we say that a−1 is invertible, if there exists an element
bÎS such that a* b = b * a = e
Also, in this case, b is called the inverse of a and we write, a−1 = b
Addition on N has no identity element and accordingly N has no
invertible element.
Multiplication on N has 1 as the identity element and no element
other than 1 is invertible.
Let S be a finite set containing n elements.
Then, the total number of binary operations on S in nn 2
.
Let S be a finite set containing n elements.
Then, the total number of commutative binary operation on S is
n.n (n +1)/2 

In a Venn diagram, the universal set is represented by a rectangular region and a set is represented by circle or a closed geometrical figure inside the universal set.