What is the Equivalence of Biconditional Connectives?

[Herricane]

Homework Statement

Show that P <-> Q is equivalent to (P ^ Q) V (¬P ^ ¬Q)

Homework Equations

P <-> Q is equivalent to (P -> Q) ^ (Q -> P)
P -> Q is equivalent to ¬p V Q
P -> Q is equivalent to ¬(P ^ ¬Q)
p -> Q is equivalent to ¬Q -> ¬P

The Attempt at a Solution

P <-> Q
(P -> Q) ^ (Q -> P)
(¬P V Q) ^ (¬Q V P)

I stuck I can't use the distributive law because I don't have one that is common to both parts.

P <-> Q
(P -> Q) ^ (Q ->P)
(¬Q -> ¬P)^(Q -> P) contrapositve law
(Q v ¬P) ^ (¬Q V P)
again I can't use the distributive law

Can anyone give my any hints?

Thanks

 

[LCKurtz]

Why not just make a truth table? It only has 4 rows.

 

[Herricane]

I was thinking of doing that, but I just thought there was a way to do it with the laws.

 

[honestrosewater]

If the problem is to use the laws, don't you have to use the laws?

Why can't you use distribution? Distribute the whole left (or right) conjunct.

(¬P V Q) ^ (¬Q V P)
((¬P V Q) ^ ¬Q) V ((¬P V Q) ^ P)

And use it again.

((¬P ^ ¬Q) V (Q ^ ¬Q)) V ((¬P ^ P) V (Q ^ P))

What's wrong with (Q ^ ¬Q)?