# What is Antisymmetric Relation – Definition and Examples

Here you will learn what is antisymmetric relation on sets with definition and examples.

Let’s begin –

## What is Antisymmetric Relation ?

Definition : Let A be any set. A relation R on set A is said to be an antisymmetric relation iff

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

Note :

(i)  It follows from this definition that if (a, b) $$\in$$ R but (b, a) $$\notin$$ R, then also R is an antisymmetric relation.

(ii)  The identity relation on a set A is an antisymmetric relation.

Given below are some antisymmetric relation examples.

Example : Let R be a relation on the set N of natural numbers defined by

x R y  $$\iff$$  ‘x divides y’  for all x, y $$\in$$ N.

This relation is an antisymmetric relation on N. Since for any two numbers a, b $$\in$$ N.

a | b and b | a $$\implies$$ a = b i.e. a R b and b R a $$\implies$$ a = b

It should be noted that this relation is not antisymmetric on the set Z of integers, because we find that for any non-zero integer a, a R (-a) and (-a) R a but a $$\ne$$ – a.

Example : The relation $$\le$$ (“less than or equal to”) on the set R of real numbers is antisymmetric, because a $$\le$$ b and b $$\le$$ a $$\implies$$ a = b for all a, b $$\in$$ R.