Give an example of a relation which is reflexive and symmetric bu… - Vidyadip

Question

Give an example of a relation which is reflexive and symmetric but not transitive.

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Answer

Let us take A = {2,4,6}
Define a relation R on A as:
A = {(2,2), (4,4), (6,6), (2,4), (4,2), (4,6), (6,4)}
Relation of R is reflexive as for every a $\in$ A,
(a,a) $\in$ R
$\Rightarrow$ (2,2), (4,4), (6,6) $\in$ R,
Relation R is symmetric as (a,b) $\in$ R
$\Rightarrow$ (b,a) $\in$ R for all a ,b $\in$ R
And Relation R is not transitive as (2,4), (4,6) $\in$ R,
but (2,6) $\notin$ R
Therefore, relation R is reflexive and symmetric but not transitive.

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