To prove , begin with and derive more facts until you reach . Often, direct proofs apply to questions of the form because the various s together can lead to intermediate logical steps.
Since an implication and its contrapositive are equivalent, you can prove the contrapositive instead. Proof by contraposition can be useful for questions of the form for the same reason as above because this is equivalent to .
To prove , assume is false and that results in something known to be false . Proof by contradiction is a type of proof by contraposition because is equivalent to .
Suppose there are cases of which at least one must be true. You can prove by showing is true for each of the when is true. This is because . Proof by cases is helpful when knowing gives you information to complete the proof that alone does not.