# To prove equivalence relation

1. May 25, 2009

### soopo

1. The problem statement, all variables and given/known data

We have an equivalence relation such that
A <-> B.

Prove that the equivalence relation is true.

3. The attempt at a solution

Let
P: A -> B
Q: B -> A

Let's prove the relation by contradiction.
Assume
$$\neg A -> \neg B$$

The previous assumption is the same as Q. Thus, we have a contradiction, since
it is impossible that both of the following Q and $$\neg Q$$
are true at the same time, where

Q: B -> A and
$$\neg Q: \neg A -> \neg B$$ which is the same as B -> A.

Thus, the equivalence relation is true between A and B.

2. May 25, 2009

### qntty

Is A <-> B given or is that what you're trying to prove?

3. May 25, 2009

### AUMathTutor

Are you trying to show that <-> is an equivalence relation on propositions?

If that is the case, you have to show the following three things:

A <-> A for all A.
if A <-> B then B <-> A.
if A <-> B AND B <-> C then A <-> C.

How should you show these? I'd probably use truth tables and the definition if <-> in terms of -> and "AND".