Prove that for all integers m and n,m-n is even if, and only if, both m and n are even or both m and n are odd.

To prove:

That for all integers m and n, m - n is even if, and only if, both m and n are even or both m and n are odd.

Proof:

Let us consider m,n∈ℤ where ℤ is the set of integers.

⇒m−n is even

Case I:

Suppose m is even and n is odd, then

m=2q,n=2p+1, p,q∈ℤm−n=2q−(2p+1)=2q−2p−1=2(q−p)−1=2k−1, k∈ℤ

Which is odd, and this is a contradiction.

Case II:

Suppose m is odd and n is even, then

m=2q+1,n=2p, p,q∈ℤm−n=2q+1−2p=2