Check whether the following statement is true | vedclass

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(A) Let the statements be defined as follows:$p: xy$ is odd.$q:$ Both $x$ and $y$ are odd.We need to verify the implication $p \Rightarrow q$ by check

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(A) Let the statements be defined as follows:$p: xy$ is odd.$q:$ Both $x$ and $y$ are odd.We need to verify the implication $p \Rightarrow q$ by checking its contrapositive statement,which is $\sim q \Rightarrow \sim p$.$\sim q:$ It is false that both $x$ and $y$ are odd. This means at least one of $x$ or $y$ is even.Let $x = 2n$ for some integer $n \in \mathbb{Z}$.Then $xy = (2n)y = 2(ny)$,which is an even number.Thus,$\sim p$ is true (i.e.,$xy$ is even).Since $\sim q \Rightarrow \sim p$ is true,the original statement $p \Rightarrow q$ is true.

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.

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