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 RA = {(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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