This document discusses indirect proof and proof by contrapositive. It shows that if n is even, then 3n + 1 is odd by substituting n = 2k and using properties of even and odd numbers. It also proves the contrapositive: if 3n + 1 is even, then n must be odd, by assuming n is not odd and reaching a contradiction.

1. INDIRECT PROOF AND PROOF BY
CONTRAPOSITIVE
Prepared by: Ma. Irene Gonzales and Min Young Park

2. 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.
CONTRAPOSITIVE
If 3n + 1is even, then n is odd.

3. 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.
INDIRECTPROOF
If 3n + 1is even, then n is odd.