How To Prove Conditional Statement

Outline for Proof by Inductiion

Proposition. The statements S1, S2, S3, S4,... are all true

Proof. (Induction)
(1) Prove that the first statement S1 is true.
(2) Given any integer K > 1, prove that the statement Sk => Sk1+1 is true. It follows by mathematical induction that every Sn is true.

Outline for Direct Proof

Proposition if P. then Q.

Proof. Suppose P.
.
.
.
Therefore Q.

Example :

If X is odd, then X^2 is odd.

Suppose X is odd.

Then, X = 2a + 1
Thus, X^2 = ( 2a + 1 ) ^2
= 4a^2 + 4a + 1
= 2 ( 2a^2 + 2a) + 1
= 2b + 1 , b = 2a^2 + 2a, a ∈ Z

Contradiction

Outline for Proof by Contradiction

Proposition P.

Proof. Suppose ~ P
.
.
.
Therefore C^ ~C

Example :

Prove that if n is an integer and 3n+2 is even,
then n is even using a contradiction

For the sake of contradiction,

Suppose 3n+2 is even, and n is odd

n = 2a+1
3n+2 = 3 (2a+1) + 2
= 6a+3+2
= 6a+5
= 2 (3a+2) + 1
= 2b+1 , b = 3a +2, a ∈ Z


From the last line, 3n+2 is odd but previously we deduced that 3n+2 is even.

It is contradiction.

Therefore, the given preposition is true.

Contrapositive

Outline for Contrapositive Proof

Proposition if P, then Q.

Proof. Suppose ~ Q
.
.
.
Therefore ~ P

Example :

If X is odd, then X^2 is