# Types of Relations on the Set A

Let A be a set and R is a relation on A, i.e. R A × A. Then we define

1. Void Relation : If R = φ , then R is called a void relation on A.
2. Universal Relation : If R = A×A, then R is called an universal relation on A.
3. Identity Relation : A relation R is defined as an identity relation if R = {(a, a) : a  A}. Thus in an identity relation on A, every element of A is related to itself only. Identity relation on A is also denoted by IA. Thus
IA = {(a, a) : a A}
Example If A = {1, 2, 3}, then IA = {(1, 1), (2, 2), (3,3)}
4. Reflexive Relation : A relation R is said to be a reflexive relation on A if every element of A is related to itself.
Thus R is reflexive (a,a) R, i.e. aRaaA
[The symbolis read as “for every element”]
Example : Let A = {1, 2, 3} be a set.
Then R = {(1, 1), (2, 2), (3, 3), (1, 3), (2, 1)} is a reflexive relation on A.
5. Symmetric Relation : A relation R on a set A is defined as a symmetric relation if (a,b) R (b, a) R That is, aRb bRa, where a, bA.
Example : Let A = {1, 2, 3, 4} and let R1 be relation on
A given by R1 = {(1, 3), (1, 4), (3, 1), (2, 2), (4, 1)} is a symmetric relation on A.
6. Transitive relation : A relation R on a set A is defined as a transitive relation if (a,b) R and (b,c) R (a,c) R
That is, aRb and bRc aRc, where a, b, c, A.
Example : Let L be the set of all straight line in a plane. Then the relation ‘is parallel to on L is a transitive relation, because of any 123 L.
1||2 and 2||3  1||3
7. Antisymmetric Relation : A relation R on a set A is antisymmetric if (a,b) R and (b,a) R a = b
If (a, b)  R and (b, a) R, then still R is an antisymmetric relation.
Example : Let R be a relation on the set N of natural numbers defined by
xRY  ‘x divides y’ for all x, y  N
This relation is a antisymmetric relation on N.
Since for any two numbers a, b  N.
a/b and b/a  a = b i.e. aRb and bRa  a = b