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.

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