Question

Find the truth value of the following statement : "If x, y $\in Z$ are such that x and y are odd, then xy is odd."

• A. True
• B. False

Answer 1

The correct option is A

True

This is an implication. This can be validated by 2 methods.

Method 1: Direct Method.

In this we assume that p is true and will try to find out the truth value of q.

Here $p:x,y\in Z$ and $x$ and $y$ are odd

$q:xy$ is odd

Since p is true $x$, $y$ can be represented as $2m+1$ and $2n+1$.

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

$=4mn+2(m+n)+1$

= Even number + 1 = odd number

i.e., q is true.

$\therefore$ Statement is validated as True.

Method 2: Contrapositive method.

In this we assume $q$ is not true and try to find out the truth value of $p$.

i.e., $\sim q\to \sim p$.

Since $q$ is not true

$xy$ = Even number

This is possible only when either of the two numbers or both the numbers are even. $\therefore$ $p$ is not true.

Hence the statement is validated as True.