What is Symmetric Relation – Definition and Examples

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

Let’s begin –

What is Symmetric Relation ?

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

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

i.e. a R b  $$\implies$$  b R a for all a, b $$\in$$ A.

Note :

(i) The identity and the universal relations on a non-void set are symmetric relations.

(ii) A relation R on the set A is not a symmetric relation if there are at least two elements a, b $$\in$$ A such that (a, b) $$\in$$ R but (b, a) $$\in$$ R.

Also Read : Types of Relations in Math

Given below are some symmetric relation examples.

Example : Let A = {1, 2, 3, 4} and let $$R_1$$ and $$R_2$$ be relations on A given by $$R_1$$ = {(1, 3),(1, 4),(3, 1),(2, 2),(4, 1)} and $$R_2$$ = {(1, 1),(2, 2),(3, 3),(1, 3)}. Clearly, $$R_1$$ is a symmetric relation on A. However, $$R_2$$ is not so, because (1, 3) $$\in$$ $$R_2$$ but (3, 1) $$\notin$$ $$R_2$$.

Example : Let S be a non-void set and R be a relation defined on power set P(S) by (A, B) $$\in$$ R $$\iff$$ A $$\subseteq$$ B for all A, B $$\in$$ P(S). Then, R is not a symmetric relation.

Note : A reflexive relation on a set A is not necessarily symmetric. For example, the relation R = {(1, 1),(2, 2),(3, 3),(1, 3)} is a reflexive relation on set A = {1, 2, 3} but it is not symmetric.