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Question:
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Amongs the given statements below
___________ is a tautology
Options:
___________ is a tautology
Solution:
1117 Upvotes
Verified Answer
The correct answer is:
$(\sim p \vee \sim q) \vee(p \vee \sim q)$
\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.
\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.
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