Here you will learn what is symmetric relation on sets with definition and examples.
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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 \(\in\) Aย
i.e. a R bย \(\implies\)ย 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.
