Question

Let $p, \, q$ and $r$ be three statements, then $\left(\sim p \rightarrow q\right) \rightarrow r$ is equivalent to

• A. $\left(\sim p \lor r\right)\land \left(q \lor r\right)$
• B. $\left(p \rightarrow r\right)\land \left(q \rightarrow r\right)$
• C. $\left(\text{~} \textit{p} \land r\right)\lor \left(q \lor r\right)$
• D. $\left(p \rightarrow q\right) \rightarrow r$

Answer 1

Answer: (B)

$\left(\sim p \rightarrow q\right) \rightarrow r\equiv \left(p \lor q\right) \rightarrow r$
$\equiv \sim \left(p \lor q\right)\lor r$
$\equiv \left(\sim p \land \sim q\right)\lor r$
$\equiv \left(\sim p \lor r\right)\land \left(\sim q \lor r\right)\equiv \left(p \rightarrow r\right)\land \left(q \rightarrow r\right)$