GATE 2016 CS – Question 11
Let $p, q, r, s$ represent the following propositions.
$p$: $x \in \{8, 9, 10, 11, 12\}$
$q$: $x$ is a composite number
$r$: $x$ is a perfect square
$s$: $x$ is a prime number
The integer $x \geq 2$ which satisfies $\neg((p \Rightarrow q) \wedge (\neg r \vee \neg s))$ is ________.
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Correct answer: 11
Explanation
The statement is true exactly when $(p \Rightarrow q) \wedge (\neg r \vee \neg s)$ is false. The part $\neg r \vee \neg s$ is always true, because no number is both a perfect square and a prime. So we need $p \Rightarrow q$ to be false, which means $p$ is true and $q$ is false. That is, $x$ is in $\{8, 9, 10, 11, 12\}$ and is not composite. The only such number is 11.