MHT CET · Maths · Mathematical Reasoning
If the statements \(p, q\) and \(r\) have the truth values \(F, T, F\) respectively, then the truth values of the statement patterns \((p \wedge \sim q) \rightarrow r\) and \((p \vee q) \rightarrow r\) are respectively
- A \(\mathrm{T}, \mathrm{T}\)
- B \(\mathrm{T}, \mathrm{F}\)
- C \(\mathrm{F}, \mathrm{T}\)
- D F, F
Answer & Solution
Correct Answer
(B) \(\mathrm{T}, \mathrm{F}\)
Step-by-step Solution
Detailed explanation
\(\begin{aligned}
& (p \wedge \sim q) \rightarrow r \\
& \equiv(F \wedge \sim T) \rightarrow F \\
& \equiv(F \wedge F) \rightarrow F \\
& \equiv F \rightarrow F \\
& \equiv T
\end{aligned}\)
Consider \((p \vee q) \rightarrow r\)
\(\begin{aligned}
& \equiv(\mathrm{F} \vee \mathrm{~T}) \rightarrow \mathrm{F} \\
& \equiv \mathrm{~T} \rightarrow \mathrm{~F} \\
& \equiv \mathrm{~F}
\end{aligned}\)
Truth values of \((p \wedge \sim q) \rightarrow r\) and \((p \vee q) \rightarrow r\) are T and F respectively.
& (p \wedge \sim q) \rightarrow r \\
& \equiv(F \wedge \sim T) \rightarrow F \\
& \equiv(F \wedge F) \rightarrow F \\
& \equiv F \rightarrow F \\
& \equiv T
\end{aligned}\)
Consider \((p \vee q) \rightarrow r\)
\(\begin{aligned}
& \equiv(\mathrm{F} \vee \mathrm{~T}) \rightarrow \mathrm{F} \\
& \equiv \mathrm{~T} \rightarrow \mathrm{~F} \\
& \equiv \mathrm{~F}
\end{aligned}\)
Truth values of \((p \wedge \sim q) \rightarrow r\) and \((p \vee q) \rightarrow r\) are T and F respectively.
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