Show work 2 Prove the following statement in two ways a usin

2. Prove the following statement in two ways, (a using contradiction and (b) using contraposition. Note this means that you need to write TWO proofs. \"For all integers mand n, if mn is even, then mis even or nis even

Solution

(a) Proof using contradiction

Assume both m and n are odd

But mn is even but product of two odd numbers is odd.

Hence a contradiction.

Hence our assumption that both m,n are odd is wrong,

So either m or n must be even or both m,n must be even

(b) Proof by contraposition

The contrapositive of this statement is:

If m,n are both odd then mn is odd for all integers m,n

m,n are given as odd integers. Product of two odd integers is odd.

Hence, m,n are odd.

Hence proved.