Question

For statement "If $x, y \in Z$ such that $x y$ is odd, then both $x$ and $y$ are odd", which of the following is a correct option?

• A. The validity of above statement can be checked by contrapositive method
• B. The above statement is true
• C. The above statement is false
• D. Both (a) and (b)

Answer 1

Answer: (D)

Let us name the statements as below
$p: x y$ is odd.
$q$ : both $x$ and $y$ are odd.
We have, to check whether the statement $p \Rightarrow q$ is true or not, i.e., by checking its contrapositive statement i,e., $\sim q \Rightarrow \sim p$.
Now, $\sim q$ : it is false that both $x$ and $y$ are odd. This implies that $x($ or $y$ ) is even.
Then, $x=2 n$ for some integer $n$.
Therefore, $x y=2 n y$ for some integer $n$. This shows that $x y$ is even. That is $\sim p$ is true.
Thus, we have shown that $\sim q \Rightarrow \sim p$ and hence the given statement is true.