# For the statements p and q, consider the following compound statements : (a) $\left ( \sim q\; \wedge\left ( p\rightarrow q \right ) \right )\rightarrow \sim p$ (b) $\left ( (p\; \vee q) \wedge \sim p\right )\rightarrow q$ Then which of the following statements is correct ? Option: 1 (b) is a tautology but not (a). Option: 2 (a) is a tautology but not (b). Option: 3 (a) and (b) both are not tautologies. Option: 4 (a) and (b) both are tautologies.

$\text { (A) }(\sim \mathrm{q} \wedge(\mathrm{p} \rightarrow \mathrm{q})) \rightarrow \sim \mathrm{p}$

$\begin{array}{|c|c|c|c|c|c|c|} \hline \mathrm{p} & \mathrm{q} & \sim \mathrm{q} & \mathrm{p} \rightarrow \mathrm{q} & \sim \mathrm{p} & (\sim \mathrm{q} \wedge(\mathrm{p} \rightarrow \mathrm{q})) & \\ \hline \mathrm{T} & \mathrm{T} & \mathrm{F} & \mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{T} \\ \mathrm{T} & \mathrm{F} & \mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{F} & \mathrm{T} \\ \mathrm{F} & \mathrm{T} & \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{F} & \mathrm{T} \\ \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} \\ \hline \end{array}$

$\text { (B) }((p \vee q) \wedge \sim p) \rightarrow q$

$\begin{array}{|c|c|c|c|c|c|} \hline \mathrm{p} & \mathrm{q} & \mathrm{p} \vee \mathrm{q} & \sim \mathrm{p} & (\mathrm{p} \vee \mathrm{q}) \wedge \sim \mathrm{p} & \\ \hline \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{T} \\ \mathrm{T} & \mathrm{F} & \mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{T} \\ \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} \\ \mathrm{F} & \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{F} & \mathrm{T} \\ \hline \end{array}$

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