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

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

NTA AbhyasNTA Abhyas 2020Mathematical Reasoning

Solution:

$p\Rightarrow \left(q \Rightarrow p\right)\equiv \sim p\lor \left(\sim q \lor p\right)$
$\equiv \left(\sim p \lor p\right)\lor \left(\sim q\right)\equiv t\lor \left(\sim q\right)\equiv t$
hence, $p\Rightarrow \left(q \Rightarrow p\right)$ is a tautology
Now, $p\Rightarrow \left(p \Rightarrow q\right)$ is false when $p$ is true and $q$ is false
$p\Rightarrow \left(q \lor p\right)\equiv \sim p\lor \left(p \lor q\right)\equiv \left(\sim p \lor p\right)\lor q$
$t\lor q\equiv t$
Hence, $\left(B\right)$ is correct.
Also, $p\Rightarrow \left(p \land q\right)$ is false when $p$ is true and $q$ is false
Also, $p\Rightarrow \left(p \Leftrightarrow q\right)$ is false when $p$ is true and $q$ is false