Let be an equivalence relation defined on a nonempty set A

Let ~ be an equivalence relation defined on a nonempty set A. Define a second relation, call it #, on A, as follows: a # b means that it is NOT the case that a ~ b. Prove that # is NEVER an equivalence relation on A.

Solution

Let us assume is possible # is an equivalence relation on A.

First of all by definition of #, a not ~b.

For any two elements a, b then a#b implies b#a.

and a #a.

If a#b and b #c, these imply a#c.

This shows that a not related to b but equivalence relation.

A contradiction.

Hence # is never an equivalence relation on A.