(C) We know that the negation of an implication is given by $\sim (p \Rightarrow q) \equiv p \wedge \sim q$. Applying this to the expression in option

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$\sim (\sim p \Rightarrow \sim q) \equiv \sim p \wedge q$

(C) We know that the negation of an implication is given by $\sim (p \Rightarrow q) \equiv p \wedge \sim q$. Applying this to the expression in option $C$: $\sim (\sim p \Rightarrow \sim q) \equiv (\sim p) \wedge \sim (\sim q)$ $\equiv \sim p \wedge q$. Therefore,the statement in option $C$ is true.

Class 11 Mathematics Mathematical Reasoning Mathematical logic

Difficult

Which of the following logical equivalences is true?

A
$p$ $\Rightarrow q \equiv \sim p$ $\Rightarrow \sim q$
B
$\sim (p \Rightarrow \sim q) \equiv \sim p \wedge q$
C
$\sim (\sim p \Rightarrow \sim q) \equiv \sim p \wedge q$
D
$\sim (p \Leftrightarrow q) \equiv [\sim (p$ $\Rightarrow q) \wedge \sim (q$ $\Rightarrow p)]$