We have an equivalence relation such that
A <-> B.
Prove that the equivalence relation is true.
The Attempt at a Solution
P: A -> B
Q: B -> A
Let's prove the relation by contradiction.
[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.