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!
