Section 1.4
Basic Proof Methods1
BASIC DEFINITIONS:
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
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An integer x is odd iff x=2k+1 for somr integer
k.
A integer is even iff x=2t for some integer t.
Let a,b be integers. We say a divides b iff there
exists an integer m such that b=am.
‫مسلمات‬
Axioms or postulate : statements that are assumed to be true.
Example: It is not the case that x is even and prime.
Example: The statement x is odd can be replaced by
x=2k+1 for some integer k.
Also to prove that x is odd, it suffices to show that x
can be expressed as x=2k+1.
The tautology [P  (P  Q)]  Q is called modus ponens rule
which means that if P and P  Q are both true, we deduce that
Q must be true
.
WORKING BACKWARD
HOW TO PROOF EACH OF THE FOLLOWING CONDITIONAL SENTENCES:
p  (Q  R )
we have two parts : (1) we prove (P  Q),
(2) we prove (P  R)
( P  Q)  R
p  (Q  R )
we assume both P and Q, then deduce R
Either we prove ( P  ~Q)  R or ( P  ~ R)  Q
We have two cases :
( p  Q)  R
caes (1) Assume P, ..., therefore R
caes (2) Assume Q,..., therefore R
Proof by cases: