Which of the following statement pattern is a contradiction? $\begin{array}{l} \mathrm{S}_{1}…

Which of the following statement pattern is a contradiction? $\begin{array}{l} \mathrm{S}_{1} \equiv(\mathrm{p} \rightarrow \mathrm{q}) \wedge(\mathrm{p} \wedge \sim \mathrm{q}) \\ \mathrm{S}_{2} \equiv[\mathrm{p} \wedge(\mathrm{p} \rightarrow \mathrm{q})] \rightarrow \mathrm{q} \\ \mathrm{S}_{3} \equiv(\mathrm{p} \vee \mathrm{q}) \rightarrow \sim \mathrm{p} \\ \mathrm{S}_{4} \equiv[\mathrm{p} \wedge(\mathrm{p} \rightarrow \mathrm{q})] \leftrightarrow \mathrm{q} \end{array}$
  1. $\mathrm{S}_{4}$
  2. $\mathrm{~S}_{1}$
  3. $\mathrm{~S}_{2}$
  4. $\mathrm{~S}_{3}$

Solution

\begin{array}{|l|l|l|l|l|l|} \hline 1 & 2 & 3 & 4 & 5 & 6 \\ \hline \mathrm{p} & \mathrm{q} & \mathrm{p} \rightarrow \mathrm{q} & \sim \mathrm{q} & \mathrm{P} \wedge \sim \mathrm{q} & (\mathrm{p} \rightarrow \mathrm{g}) \wedge(\mathrm{p} \wedge \sim \mathrm{q}) \\ \hline \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{F} \\ \hline \mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{F} \\ \hline \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{F} \\ \hline \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{F} & \mathrm{F} \\ \hline \end{array} All entries in last column are $\mathrm{F}$. So $\mathrm{S}_{1}$ is a contradiction. This problem can be alternatively solved as follows : $\begin{aligned} S_{1} & \equiv(p \rightarrow q) \wedge(p \wedge \sim q) \\ & \equiv[(\sim p \vee q)] \wedge[\sim(\sim p \vee q)] \\ & \equiv F \end{aligned}$

Asked in: MHT CET 2020 (12 Oct Shift 2)

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