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  • For the statements p and q, consider the following compound statements : (a) <img alt="\left ( \sim q\; \wedge\left ( p\rightarrow q \right ) \right )\rightarrow \sim p" src="https://entrancecorner.codecogs.com/gif.latex?%5Cleft%20%28%20%5Csim%20q%5C%3

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.

Answers (1)

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\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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Suraj Bhandari

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