Methods of Proof A mathematical theorem is usually of the form pq where p is called hypothesis or premise, and q is called conclusion. p is often of the form p1p2…pn If pq is a tautology, then q logically follows from To ‘prove the theorem’ means to show that the implication is a tautology Arguments based on tautologies represent universally correct methods of reasoning; such arguments are called rules of inference.

Indirect Proof Methods The first indirect method of proof, follows from the tautology (pq)  (~q~p), i.e. an implication is equivalent to its contrapositive The second indirect proof: by contradiction is based on the tautology (pq) ((p  ~ q)  F) To disprove the result, only to find one counterexample for which the claim fails The proof of pq is logically equivalent with proving both pq and qp

Mathematical Induction To prove nn0 P(n), where n0 is some fixed integer, begin by proving the basic step: P(n0) is true and then the induction step: If P(k) is true for some kn0, then P(k+1) must also be true Then P(n) is true for all nn0 The result is called the principle of mathematical induction. In the strong form of mathematical induction, or strong induction, the induction step is to show that P(n0)P(n0+1)P(n0+2)…P(k)  P(k+1) is a tautology.