Give an example of a relation which is symmetric but neither refl… - Vidyadip

Question

Give an example of a relation which is symmetric but neither reflexive nor transitive.

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Answer

Let A = {3,4,5}
Define a relation R on A as R = {(3,4), (4,3)}
Relation R is not reflexive as (3,3), (4,4) ,(5,5) $\notin$ R.
Now, as (3,4) $\in$ R and (4,3) $\in$ R,
R is symmetric.
Further, (3,4),(4,3) $\in$ R, but (3,3) $\notin$ R
$\Rightarrow$ R is not transitive.
Therefore, relation R is symmetric but not reflexive or transitive.

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