Question

if m+n is even, then m-n is even
use an indirect proof

So I assume m-n is odd (indirect proof)
m-n=2i+1 where i is Z
then m+n = 2i +1 -2(2i+1-m)
how do i get the right side to be simplified without ms or ns? help!

Answer 1

m+n would be even if m,n both are even or both are odd, now check one by one :
even - even = even
odd - odd = even

QED

Answer 2

if m-n=2k+1

then

m+n=m-n+2n=2k+1+2n=2(k+n)+1

hence m+n is odd (a contradiction)