Proof by Cases

Some proofs need to be broken down into different cases, through which the whole proof can be evaluated. This makes proving complex proofs easier. In fact, I think we already used this method for proving equivalence. In class, we showed that an implication "P implies Q" is equivalent to its contrapositive "not Q implies not P" by showing:

P	Q	P implies Q	not Q implies not P
T	T	T	T
T	F	F	F
F	T	T	T
F	F	T	T

In each of the four cases above, we showed that P implies Q is true if and only if not Q implies not P is true. Even when P is True and Q is False, both the implication and its contrapositive are False, which means the equivalence still stand. Also note that proof by cases work in a much broader environment than propositions involving boolean variables.