Amongs the given statements below ___________ is a tautology

Amongs the given statements below ___________ is a tautology
  1. $\sim p \vee(\sim p \vee \sim q)$
  2. $\sim q \wedge(\sim p \vee \sim q)$
  3. $(\sim p \vee \sim q) \wedge(p \vee \sim q)$
  4. $(\sim p \vee \sim q) \vee(p \vee \sim q)$

Solution

\begin{array}{|c|c|c|c|c|c|c|c|c|c|} \hline \mathrm{p} & \mathrm{q} & \sim \mathrm{p} & \sim \mathrm{q} & \sim \mathrm{p} \vee \sim \mathrm{q} & (3) \vee(5) & (4) \wedge(5) & \mathrm{p} \vee \sim \mathrm{q} & (5) \wedge(8) & (5) \vee(8) \\ \hline 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \hline \mathrm{T} & \mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{F} & \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{F} & \mathrm{T} \\ \hline \mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} \\ \hline \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{F} & \mathrm{F} & \mathrm{F} & \mathrm{T} \\ \hline \mathrm{F} & \mathrm{F} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} & \mathrm{T} \\ \hline & & & & (\mathrm{a}) & (\mathrm{b}) & & (\mathrm{c}) & (\mathrm{d}) \\ \hline \end{array} All entries in column 10 are T. Hence statement(d) is a tautology.

Asked in: MHT CET 2020 (15 Oct Shift 2)

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