# Logical Equivalences Question

This is for a homework problem, so I would prefer bumps or tips in the right direction rather than full answers.

I am supposed to show the logical equivalence of $p \leftrightarrow q$ and $(p \land q) \lor (\lnot p \land \lnot q)$ using logical identities and laws. I have looked through all of the ones I could think of and have applied them different ways and had no luck, perhaps someone could offer me some pointers for what I am missing?

The laws I have tried are - commutative, associative, distributive, identity, negation, double negative, idempotent, universal bound, De Morgan's, absorption, and conditional.

Thank you!

• What is the definition of $\Leftrightarrow$ that you were given? – William Sep 12 '14 at 21:36
• I believe it is the biconditional/equivalence symbol. Is there more than one definition for this symbol? – turner Sep 12 '14 at 21:41
• All the definitions are equivalent, but how can prove anything if you do not have a fixed definition of what $\Leftrightarrow$ means? In fact, you can even define $p \Leftrightarrow q$ as $(p \wedge q) \vee (\neg p \wedge \neg q)$. – William Sep 12 '14 at 21:43

• Yep! Compare it to how you FOIL things in algebra: $$(a + b) \cdot (c + d) = (a \cdot c) + (a \cdot d) + (b \cdot c) + (b \cdot d)$$ – Adriano Sep 12 '14 at 22:06