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  4. If p and q are logical statements, then p⇒ (∼ q ⇒ p) is equivalent to

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Q. If $p$ and $q$ are logical statements, then $p\Rightarrow \left(\sim q \Rightarrow p\right)$ is equivalent to

NTA AbhyasNTA Abhyas 2022

Solution:

$p\Rightarrow \left(\sim q \Rightarrow p\right)\equiv simp\lor \left(q \lor p\right)$
$\equiv \left(sim p \lor p\right)\lor \left(q\right)\equiv t\lor \left(q\right)\equiv t$
Hence, the given statement $p\Rightarrow \left(\sim q \Rightarrow p\right)$ is a tautology
Now, $p\Rightarrow \left(p \Rightarrow q\right)$ is false when $p$ is true and $q$ is false
Hence, not a tautology
$p\Rightarrow \left(q \lor p\right)\equiv \sim p\lor \left(p \lor q\right)\equiv \left(\sim p \lor p\right)\lor q$
$t\lor q\equiv t$
Hence, $p\Rightarrow \left(q \lor p\right)$ is a tautology.
Now, $p\Rightarrow \left(p \land q\right)$ is false when $p$ is true and $q$ is false
Hence, not a tautology
Also, $p\Rightarrow \left(p \Leftrightarrow q\right)$ is false when $p$ is true and $q$ is false
Hence, not a tautology