The document explains relations in discrete mathematics, defining them as the relationships between two sets, denoted as the domain and range. It describes various types of relations, including empty, universal, identity, inverse, reflexive, symmetric, transitive, and equivalence relations. It also notes that while all functions are relations, not all relations are functions.

1. SHRIRAMSWAROOP MEMORIAL
UNIVERSITY
DISCRETE MATHEMATICS
(MMA1010)
MCA – 1st Semester
TOPIC: Relation and its types
SUBMITTED TO: SUBMITTED BY:
Dr. Ajay Singh Mohd. Shahil Alam
202310101050006

2. RELATION
 Relation in Mathematics is defined as the relationship
between two sets. If we are given two sets set A and set
B and set A has a relation with set B then each value of
set A is related to a value of set B through some unique
relation. Here, set A is called the domain of the relation,
and set B is called the range of the relation.
 For example if we are given two sets, Set A = {1, 2, 3, 4}
and Set B = {1, 4, 9, 16} then the ordered pair {(1, 1), (2,
4), (3, 9), (4, 16)} represents the relation defined as, R,
A: → B {(x, y): y = x2: y ϵ B, x ϵ A}.

3. TYPES OF RELATIONS
 Empty relation: A relation will be known as
empty relation if the elements of set have no
relation with each other.
Example:- R = Ø
 Universal Relation: In this relation, every
element of set A has a relation with every
element of set B. That's why this relation is also
called a universal relation, and the universal
relation and empty relation are also known as
the trivial relation.

4.  Identity Relation: In the identity relation, every
element of A has a relation to itself only. The
identity relation is described as follows:
A : A → A
 Inverse Relation: Assume that there are two
sets, set A and set B, and they have a relation
from A to B. This relation will be described as
R∈ A × B. The inverse relation will be obtained
when we replace the first element of each pair
with the second element in a set. The inverse
relation is described as follows:
R -1 = {(b, a) : (a, b) ∈ R}

5.  Reflexive Relation: A relation will be known as
reflexive relative if every element of set A is
related to itself. The word reflexive means that
in a set, the image of every element has its own
reflection.
 Symmetric Relation: Suppose there are two
elements in a set A, i.e., a, b. The relation R on
a set A will be called symmetric relation if
element 'a' has relation with 'b' is true, then 'b'
has a relation with 'a' will also be true. The
symmetric relation is described as follows:
If (a, b) ∈ R then (b, a) ∈ R, for all a and b ∈ A

6.  Transitive Relation: Suppose there are three
elements in a set A, i.e., a, b, and c. The relation R
on a set A will be known as transitive relation if 'a'
has a relation with 'b' and 'b' has a relation with 'c',
then 'a' will be also has a relation with 'c'. The
transitive relation is described as follows:
If (a, b) ∈ R , (b, c) ∈ R , then (a, c) ∈ R for all a, b, c
∈ A
 Equivalence Relation: A relation will be known as
equivalence relation if it satisfies the properties of
symmetric, transitive, and reflexive.
Note: All relations are not functions, but all functions
are relations.