Answer
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Hint – A tautology is a statement that is always true, no matter what. If you construct a truth table for a statement and all of the column values for the statement are true (T), then the statement is a tautology because it's always true!
Complete step-by-step answer:
$
\left( i \right)\;(p \vee q) \vee (p \vee \sim q) = p \vee (q \vee \sim q) = p \vee t = t \\
\left( {ii} \right)\;(p \wedge q) \wedge (p \vee \sim q) \\
\left( {iii} \right)\;(p \vee q) \wedge (p \vee \sim q) = p \vee (q \wedge \sim q) - p \vee f = p \\
\left( {iv} \right)\;(p \vee q) \vee (p \wedge \sim q) \\
$
\[\begin{array}{*{20}{l}}
p&q&{ \sim q}&{p \vee q}&{p \wedge \sim q}&{(p \vee q) \vee (p \wedge \sim q)} \\
T&T&F&T&F&T \\
T&F&T&T&T&T \\
F&T&F&T&F&T \\
F&F&T&F&F&F
\end{array}\]
$
(p \wedge q) \wedge (p \vee \sim q) \\
\begin{array}{*{20}{l}}
p&q&{ \sim q}&{p \wedge q}&{p \wedge \sim q}&{(p \wedge q) \wedge (p \vee \sim q)} \\
T&T&F&T&T&T \\
T&F&T&F&T&F \\
F&T&F&F&F&F \\
F&F&T&F&T&F
\end{array} \\
$
Therefore, you can see now option (a) is correct.
Note – In this problem, first we try to construct a truth table for all the given statements. After that we will see whether these statements are a tautology or not. Hence, we get the required answer.
Complete step-by-step answer:
$
\left( i \right)\;(p \vee q) \vee (p \vee \sim q) = p \vee (q \vee \sim q) = p \vee t = t \\
\left( {ii} \right)\;(p \wedge q) \wedge (p \vee \sim q) \\
\left( {iii} \right)\;(p \vee q) \wedge (p \vee \sim q) = p \vee (q \wedge \sim q) - p \vee f = p \\
\left( {iv} \right)\;(p \vee q) \vee (p \wedge \sim q) \\
$
\[\begin{array}{*{20}{l}}
p&q&{ \sim q}&{p \vee q}&{p \wedge \sim q}&{(p \vee q) \vee (p \wedge \sim q)} \\
T&T&F&T&F&T \\
T&F&T&T&T&T \\
F&T&F&T&F&T \\
F&F&T&F&F&F
\end{array}\]
$
(p \wedge q) \wedge (p \vee \sim q) \\
\begin{array}{*{20}{l}}
p&q&{ \sim q}&{p \wedge q}&{p \wedge \sim q}&{(p \wedge q) \wedge (p \vee \sim q)} \\
T&T&F&T&T&T \\
T&F&T&F&T&F \\
F&T&F&F&F&F \\
F&F&T&F&T&F
\end{array} \\
$
Therefore, you can see now option (a) is correct.
Note – In this problem, first we try to construct a truth table for all the given statements. After that we will see whether these statements are a tautology or not. Hence, we get the required answer.
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