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

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

hoblos · Answer

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

tux · Answer

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