MHT CET · Maths · Mathematical Reasoning
If \((\sim \mathrm{p} \wedge \mathrm{q}) \rightarrow \mathrm{r}\) is false then the truth values of \(\mathrm{p}, \mathrm{q}, \mathrm{r}\) are respectively
- A \(\mathbf{F}, \mathbf{T}, \mathbf{F}\)
- B \(\mathrm{F}, \mathrm{T}, \mathrm{T}\)
- C \(\mathrm{T}, \mathrm{T}, \mathrm{F}\)
- D \(\mathrm{F}, \mathrm{F}, \mathrm{T}\)
Answer & Solution
Correct Answer
(A) \(\mathbf{F}, \mathbf{T}, \mathbf{F}\)
Step-by-step Solution
Detailed explanation
Given \((\sim p \wedge q) \rightarrow r\) is false \(T \rightarrow F \equiv F\)
We know that \(T \rightarrow F=F\)
\(\therefore \sim p \wedge q=T \quad\) and \(r \equiv F\)
Also we know that \(\mathrm{T} \wedge \mathrm{T}=\mathrm{T}\)
\(\therefore \mathrm{q} \equiv \mathrm{T}\) and \(\sim \mathrm{p}=\mathrm{T} \Rightarrow \mathrm{p}=\mathrm{F}\)
We know that \(T \rightarrow F=F\)
\(\therefore \sim p \wedge q=T \quad\) and \(r \equiv F\)
Also we know that \(\mathrm{T} \wedge \mathrm{T}=\mathrm{T}\)
\(\therefore \mathrm{q} \equiv \mathrm{T}\) and \(\sim \mathrm{p}=\mathrm{T} \Rightarrow \mathrm{p}=\mathrm{F}\)
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