Types of Relations in Math – Reflexive, Symmetric

In this post, we will learn various types of relations in math on a set.

Let’s begin-

Types of Relations in Math

1). Void, Universal and Identity Relation

Void Relation : Let A be a set. Then \(\phi\) \(\subseteq\) A \(\times\) A and so it is a relation on A. This relation is called  the void or empty relation on set A.

In other words, a relation R on the set A is called void or empty relation, if no element of A is related to any element of A.

for example : Consider the relation R on set A = {1,2,3,4,5} defined by R = {(a,b) : a-b = 12}.

Universal Relation : Let A be a set. Then, A \(\times\) A \(\subseteq\) A \(\times\) A and so it is a relation on A. This relation is called universal relation on A.

for example : Consider the relation R on set A = {1,2,3,4,5,6} defined by R = {(a,b) : |a-b| \(\ge\) 0}.

Identity Relation : Let A be a set. Then, the relation \(I_A\) = {(a, a) : a \(\in\) A} on A is called the identity relation on A.

In other words, a relation \(I_A\) on A is called the identity relation if every element of A is related to itself only.

for example : If A = {1,2,3}, then the relation \(I_A\) = {(1,1),(2,2),(3,3)} is the identity relation on set A.

2). Reflexive Relation

A relation R on a set A is said to be reflexive if every element of A is related to itself.

Thus, R is reflexive \(\implies\) (a, a) \(\in\) R for all a \(\in\) R

for example : If A = {1,2,3}, then the relation R = {(1,1),(2,2),(3,3),(1,3),(2,1)} is the reflexive relation on A, But \(R_1\) = {(1,1),(3,3),(3,2),(2,1)}  is not a reflexive relation on A, because 2 \(\in\) A but (2,2) \(\notin\) \(R_1\).

3). Symmetric Relation

A relation R on a set A is said to be symmetric iff

(a,b) \(\in\) R \(\implies\) (b,a) \(\in\) R for all a,b \(\in\) A

i.e. aRb \(\implies\) bRa for all a, b \(\in\) A.

for example : If A = {1,2,3,4}, then the relation R = {(1,3),(1,4),(3,1),(2,2),(4,1)} is the symmetric relation on A, But \(R_1\) = {(1,1),(3,3),(2,2),(1,3)}  is not a symmetric relation on A, because (1,3) \(\in\) \(R_1\) but (3,1) \(\notin\) \(R_1\).

4). Transitive Relation

Let A be any set. A relation R on A is said to be transitive relation iff

(a,b) \(\in\) R and (b,c) \(\in\) R \(\implies\) (a,c) \(\in\) R for all a, b, c \(\in\) A.

i.e. aRb and bRc  \(\implies\) aRc for all a,b,c \(\in\) A.

for example : If A = {1,2,3}, then the relation R = {(1,2),(2,3),(1,3),(2,2)} is the transitive relation on A, But \(R_1\) = {(1,2),(2,3),(2,2),(1,1)}  is not a transitive relation on A, because (1,3) and (2,3) \(\in\) \(R_1\) but (1,3) \(\notin\) \(R_1\).

5). Equivalence Relation

A relation R on a set A is said to be an equivalence relation on A if it is reflexive, symmetric and transitive.

for example : If A = {1,2,3}, then the relation R = {(1,1),(2,2),(3,3),(2,1),(1,2),(2,3),(1,3).(3,2),(3,1)} is the equivalence relation on A, because {(1,1),(2,2),(3,3)} \(\in\) R hence it is reflexive, {(2,1),(1,2),(2,3),(3,2),(1,3),(3,1)} \(\in\) R  hence it is symmetric on A, {(1,2),(2,3),(1,3)} and {(1,3),(3,2),(1,2)} \(\in\) R hence it is transitive.

Therefore, it is a equivalence relation.

Hope you learnt types of relations in math, learn more concepts of relations and practice more questions to get ahead in the competition. Good luck!

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