INDIRECT PROOF AND PROOF BY CONTRAPOSITIVE

Prepared by: Ma. Irene Gonzales and Min Young Park

Contrapositive: If n is even, then 3n + 1 is odd.

Suppose n is even, then n = 2k for some integer

k. Then,

3n + 1 = 3 (2k) + 1 substitution

= 2 (3k) + 1 commutative property

Since 3n + 1 is twice another integer plus 1,

then 3n + 1 is an odd integer.

CO

NTR

AP

OSITIV

EIf 3n + 1is even, then n is odd.

Suppose that n is not odd, then n is even.

If n is even, then n = 2k for some integer k

so 3n + 1 = 3(2k) + 1 = 2(3k) + 1 by

commutative property.

Thus 3n + 1 is odd contrary to the

hypothesis.

IND

IREC

T PR

OO

FIf 3n + 1is even, then n is odd.