Q. 83.8( 4 Votes )

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

Answer :

To Prove: Every identity relation on a set is reflexive, but every reflexive relation is not identity relation.

Proof:

Let us first understand what ‘Reflexive Relation’ is and what ‘Identity Relation’ is.

Reflexive Relation: A binary relation R over a set A is reflexive if every element of X is related to itself. Formally, this may be written as ∀ x ∈ A: xRx.

Identity Relation: Let A be any set.

Then the relation R= {(x, x): x ∈ A} on A is called the identity relation on A. Thus, in an identity relation, every element is related to itself only.

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

Let R be a binary relation defined on A.

Let R_{A} = {(a, a): a ∈ A} is the identity relation on A.

Hence, every identity relation on set A is reflexive by definition.

Converse: Let A = {a, b, c} is the set.

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

R is reflexive as per definition.

[∵ (a, a) ∈ R, (b, b) ∈ R & (c, c) ∈ R]

But, (a, b) ∈ R

(c, a) ∈ R

⇒ R is not identity relation by definition.

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

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