Question

Prove: If ab is even then a or b is even.

Answer 1

As i understand, it is implied that numbers a and b must be integer(otherwise, the statement is false). So, this numbers can be even or odd. There are 4 possibilities:

1).a, b are both even

2).a, b are both odd

3).a is even, b is odd

4).a is odd, b is even

Firstly we will examine (2) condition: a and b are both odd means $a=2n-1$, $b=2k-1$ where n and k both $\in Z(+)$

Then $ab=(2n-1)(2k-1)=4nk - 2(n+k) +1$

4nk and 2(n+k) are both even, then their sum is even, then their sum plus 1 is odd. ab is odd.

We considered (2) situation and came to conclusion that: $(2)\to$ (ab is odd). That means: $\lnot$ (ab is odd) $\to$ $\lnot (2)$

$\lnot$ (ab is odd) means (ab is even)

(ab is even) $\to$ $\lnot (2)$.

So, if ab is even than at least one of the two numbers is even

Statement has been proven