- #1

- 225

- 0

## Homework Statement

We have an equivalence relation such that

A <-> B.

Prove that the equivalence relation is true.

## The Attempt at a Solution

Let

P: A -> B

Q: B -> A

Let's prove the relation by contradiction.

Assume

[tex]\neg A -> \neg B [/tex]

The previous assumption is the same as Q. Thus, we have a contradiction, since

it is impossible that both of the following Q and [tex] \neg Q[/tex]

are true at the same time, where

Q: B -> A and

[tex]\neg Q: \neg A -> \neg B [/tex] which is the same as B -> A.

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