Question

Prove that every identity relation is reflexive, but the converse may not be necessarily true.

Answer 1

Hint: Use the definition of an identity relation on a set A. Recall the definition of a reflexive relation. Hence verify that an identity relation is a reflexive relation. In order to prove that a reflexive relation may not be an identity relation think of a counter-example, i.e. a reflexive relation which is not an identity relation. Try adding an element in $A\times A$ which is not in identity relation to the set of identity relation. Argue that the so formed relation is not identity.

Complete step-by-step answer:
Consider a set A and let I be an identity relation defined over it.
A relation defined over a set is set to be an identity relation of it maps every element of A to itself and only to itself, i.e.
$\forall a\in A,\left( a,a \right)\in I$ and if $\left( a,b \right)\in I$ then $a=b$
Reflexive relation: A relation R defined over a set A is said to be reflexive if and only if $\forall a\in A\Rightarrow \left( a,a \right)\in R$.
Since $\forall a\in A,\left( a,a \right)\in I$, we have and I is reflexive.
Hence every identity relation is a reflexive relation.
Now consider a set A ={1,2,3} and let R ={(1,1),(2,2),(3,3),(1,3)} be a relation defined over the set A.
Since $\forall a\in A,\left( a,a \right)\in R$, R is a reflexive relation. However, since $\left( 1,3 \right)\in R$ and $1\ne 3$, we have R is not an identity relation over A. Hence a reflexive relation may not be an identity relation. Hence proved.

Note: In the questions of the above type, we need to understand and remember the definitions of various types of identity relation. In order to prove that a statement is correct, we need to come up with formal proof, and in order to prove that a statement is wrong, we need to come up with a counter-example, as done above.