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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) R (b, a) R for all a, b

i.e. a R b    b R a for all a, b 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 A such that (a, b) R but (b, a) R.

Also Read : Types of Relations in Math

Given below are some symmetric relation examples.

Example : Let A = {1, 2, 3, 4} and let R1 and R2 be relations on A given by R1 = {(1, 3),(1, 4),(3, 1),(2, 2),(4, 1)} and R2 = {(1, 1),(2, 2),(3, 3),(1, 3)}. Clearly, R1 is a symmetric relation on A. However, R2 is not so, because (1, 3) R2 but (3, 1) R2.

Example : Let S be a non-void set and R be a relation defined on power set P(S) by (A, B) R A B for all A, B 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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