Question

Check whether the following statement is true or not.
If $x,y\in Z$ are such that $x\text{and}y$ are odd, then $xy$ is odd.

Answer 1

Given Statement: If $x,y\in Z$ are such that $x\text{and}y$ are odd, then $xy$ is odd

Let $p:x,y\in Z$ such that $x\text{and}y$ are odd.
$q:xy$ is odd.

So, the given statement is of the form $p\Rightarrow q$

Using direct method,
By assuming that $p$ is true, prove that $q$ must be true. i.e.,

Assuming $x\&y$ are odd, prove that $xy$ is odd.

$\because x\&y$ are odd,

Let $x=2m+1andy=2n+1$ Where $m,n\in Z$

Thus,
$xy=(2m+1)(2n+1)$

$\Rightarrow xy=2m(2n+1)+1(2n+1)$

$\Rightarrow xy=\left(2m\right)\left(2n\right)+2m+2n+1$

$\Rightarrow xy=4mn+2m+2n+1$

$\Rightarrow xy=2(2mn+m+n)+1$

This shows that $xy$ is odd.

Hence, $q$ is true.

Therefore, the given statement is true.