Q.Check whether the following statement is true or false by proving its contrapositive.
If x, y $\in \mathbb{Z}$ such that xy is odd, then both x and y are odd.

Statement $p$: If $xy$ is odd, then both $x$ and $y$ are odd.

The contrapositive of $p$ is: "If $x$ and $y$ are not both odd (i.e., at least one of $x, y$ is even), then $xy$ is not odd (i.e., $xy$ is even)."

Proof of the contrapositive: Suppose, without loss of generality, that $x$ is even. Then $x = 2k$ for some integer $k$. Therefore:

Since $ky$ is an integer, $xy$ is of the form $2 \times (\text{integer})$, i.e., $xy$ is even. (The case where $y$ is even is symmetric.) …

To prove p ⇒ q by contrapositive: assume ¬q (here, x or y is even) and derive ¬p (xy is even). If the contrapositive holds, the o …

• Writing the contrapositive incorrectly (e.g., negating only "both x and y are odd" as "both x and y are even" instead of the correct "at least one is even"). …