The negation of the statement "The number is an odd number if and only if it is divisible by $3$ . "
The negation of the statement "The number is an odd number if and only if it is divisible by $3$ . "
(A) The number is an odd number but not divisible by 3 or the number is divisible by 3 but not odd.
The number is not an odd number iff it is not divisible by 3 .
The number is not an odd number but it is divisible by 3 .
The number is not an odd number or is not divisible by 3 but the number is divisible by $3$ or odd.
Solution
Let $\mathrm{p}$ : The number is an odd number,
$\mathrm{q}:$ The number is divisible by 3
$\sim(p \leftrightarrow q) \equiv(p \wedge \sim q) \vee(q \wedge \sim p)$
$\therefore \quad$ The negation of the given statement is "The number is an odd number but not divisible by 3 or the number is divisible by 3 but not odd'.