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Joint Entrance Examination
Mathematics
Mathematical Reasoning
Question 5
Question 5 of 103
Medium
MCQ
The statement \[\sim[p \vee(\sim(p \wedge q))]\] is equivalent to :
A
\[p \to \left( {p \leftrightarrow q} \right)\]
B
\[p \to \left( {p \to q} \right)\]
C
\[p \to \left( {p \vee q} \right)\]
D
\[p \to \left( {p \wedge q} \right)\]
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Official solution
\[ \[
\begin{array}{|ccccc|}\\ p & q & p \wedge q & \sim(p \wedge q) & \begin{array}{c} p \vee \sim \\ (p \wedge q)\end{array}
\&
∼
[
p
∨
∼
(
p
∧
q
)
]
\begin{array}{|c|}\sim[p \vee \sim \\ (p \wedge q)]\end{array}
∼
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p
∨
∼
(
p
∧
q
)]
\& p \vee q \&
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∨
q
)
\begin{array}{|c|}\sim(p \\ \vee q)\end{array}
∼
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∨
q
)
\&
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∧
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)
∧
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\begin{array}{|c|}(p \wedge q) \wedge \\ \quad(\sim p)\end{array}
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∧
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\\ \\ \mathrm{T} \& \mathrm{T} \& \mathrm{T} \& \mathrm{F} \& \mathrm{T} \& \mathrm{F} \& \mathrm{T} \& \mathrm{F} \& \mathrm{F} \\ \\ \mathrm{T} \& \mathrm{F} \& \mathrm{F} \& \mathrm{T} \& \mathrm{T} \& \mathrm{F} \& \mathrm{T} \& \mathrm{F} \& \mathrm{F} \\ \\ \mathrm{F} \& \mathrm{T} \& \mathrm{F} \& \mathrm{T} \& \mathrm{T} \& \mathrm{F} \& \mathrm{T} \& \mathrm{F} \& \mathrm{F} \\ \\ \mathrm{F} \& \mathrm{F} \& \mathrm{F} \& \mathrm{T} \& \mathrm{T} \& \mathrm{F} \& \mathrm{F} \& \mathrm{T} \& \mathrm{F} \\ \\ \end{array}\] \] \[ \therefore \sim[p \vee(\sim(p \wedge q))]=\sim p \wedge(p \wedge q) \]
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