Define Symmetric relations

A relation $R$ in a set $A$ is called symmetric if $(a,b) \in R$ implies that $(b, a) \in R$ $\forall a, b \in A$.


Consider, for example, the set $A$ of natural numbers. If a relation A be defined by $x + y = 10$, then this relation is symmetric in $A$, for $a + b = 10$ ⇒ $b + a = 10$


But in the set $A$ of natural numbers if the relation $R$ be defined as ‘$x$ is a divisor of $y$’, then the relation $R$ is not symmetric as 3R9 does not imply 9R3; for, 3 divides 9 but 9 does not divide 3.



For a symmetric relation $R$, $R^{−1}$ = $R$.