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The negation of the statement "The number is an odd number if and only if it is divisible by $3$ . "
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The correct answer is:
(A) The number is an odd number but not divisible by 3 or the number is divisible by 3 but not odd.
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'.
$\mathrm{q}:$ 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'.
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