Question

Discrete math

Prove directly
For n∈N, if n is odd, then 3n+1 is even
For x,y∈R, if x and y are rational, then x+2y is rational

Prove by contrapositive
For n ∈N if n^3 is odd, then n is odd

Answer 1

first one..
n is odd so it can be written as 2k+1
3n+1 = 3(2k+1) +1
= 6k +4
= 2(3k+2) and this is even for any k

Answer 2

contrapositive:

n=2k
(2k)^3=8k^3=2*(4k^3)