Prove that every identity relation on a set is reflexive, but the… - Vidyadip

Question

Prove that every identity relation on a set is reflexive, but the converse is not necessarily true.

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Answer

Let A be a set.

Then $\text{I}_\text{A}=\{(\text{a, a});\text{ a}\in\text{A}\}$ is the identity relation on A.

Hence, every identity relation on a set is reflexive by defination.

Converse:

Let A = {(a, b, c)} be a set.

Let R = {(a, a), (b, b), (c, c), (a, b)} be a relation defined on A.

Clearly R is reflexive on set A, but it is not identity relation on set A as $(\text{a, b})\in\text{R}$

Hence, a reflexive relation need not be identity relation.

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