A mathematical theorem is usually of the form

1
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.

2
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

3
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(k1) 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(n01)?P(n02)?P(k) ? P(k1) is a
  tautology.