Transcript Methods of Proof - University of Macau

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.