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  4. The logical proposition (∼ (∼ p ∨ q)) ∨ (p ∧ r)∧ (∼ q ∧ r) is equivalent to

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Q. The logical proposition $\left(\sim \left(\sim p \lor q\right)\right) \lor \left(p \land r\right)\land \left(\sim q \land r\right)$ is equivalent to

NTA AbhyasNTA Abhyas 2022

Solution:

$\left[\sim \left(\sim p \lor q\right) \lor \left(p \land r\right)\right]\land \left(\sim q \land r\right)$
$=\left[\left(p \land \sim q\right) \lor \left(p \land r\right)\right]\land \left(\sim q \land r\right)$
By distributive law
$=p\land \left(\sim q \lor r\right)\land \left(\sim q \land r\right)$
Solution
undefined $p\land \sim q\land r$
$=p\land \sim q\land r$
By Associative law
$=\left(p \land r\right)\land \sim q$