Suppose A is a set Show that i A is the only relation on A t

Suppose A is a set. Show that i _A is the only relation on A that is both an
equivalence relation on A and also a function from A to A.

Solution

And let R be an equivalence relation on R

And, a,b be two distinct elements in R so that aRb

But R is equivalence hence reflexive so aRa

But R is also a function so a cannot be mapped to two distinct elements.

But we have aRa and aRb ie a is mapped to a and b which is a contradiction

Hence, no two distinct elements are related.

Hence, R is i_A is identity function on A.