P if and only if Q means, (P then q) AND (q then p)

I was asked to state that the claim is true or false, I must give a prove to say it is true and counter example if it is false.

However I say it is True;This is a bi-conditional statement which mean p if and only if q. p implies q and q implies p which means it is true when both are true or both are false. This also means we combine two conditional statements together and the above claim sates that P if and only if q, means (p then q) AND (q then p) is putting or combining two conditional statements, which is true.

Is my reasoning correct or there is a law to prove that it is indeed true?

1 Answer

I would express the proof as follows:

P if and only if Q means: P if Q and P only if Q.

By definition, P only if Q means: If not Q then not P.

So the original statement can be written as: P if Q and if not Q then not P.

P if Q means: If Q then P.

And by the contrapositive, if not Q then not P means: If P then Q.

So we are left with: Q then P and P then Q.