Question:
If \(\{(\mathrm{p} \wedge \sim \mathrm{q}) \wedge(\mathrm{p} \wedge \mathrm{r})\} \rightarrow \sim \mathrm{p} \vee \mathrm{q}\) has truth value false then truth values of the statements \(p, q, r\) are respectively
If \(\{(\mathrm{p} \wedge \sim \mathrm{q}) \wedge(\mathrm{p} \wedge \mathrm{r})\} \rightarrow \sim \mathrm{p} \vee \mathrm{q}\) has truth value false then truth values of the statements \(p, q, r\) are respectively
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An implication is false only when the antecedent is true and the consequent is false.
Updated On: Nov 13, 2026
- \(\mathrm{T},\mathrm{T},\mathrm{T}\)
- \(\mathrm{F},\mathrm{F},\mathrm{F}\)
- \(F,F,T\)
- \(T,F,T\)
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The Correct Option is D
Solution and Explanation
Step 1: Understanding the Concept
\(P\rightarrow Q\) is false only when \(P\) is true and \(Q\) is false.
Step 2: Consequent False
\(\sim p\vee q\) is false only if \(\sim p\) and \(q\) are both false.
So \(p=\text{T}\) and \(q=\text{F}\).
Step 3: Antecedent True
\((p\wedge\sim q)\wedge(p\wedge r)\) is true only if \(p\), \(\sim q\) and \(r\) are all true.
With \(p=\text{T}\), \(\sim q=\text{T}\), we need \(r=\text{T}\).
Step 4: Compare the options
So \(p,q,r\) are T, F, T, which is the last option.
Final Answer:
The truth values of p, q, r are T, F, T. \[ \boxed{\text{(D) }\text{T, F, T}} \]
\(P\rightarrow Q\) is false only when \(P\) is true and \(Q\) is false.
Step 2: Consequent False
\(\sim p\vee q\) is false only if \(\sim p\) and \(q\) are both false.
So \(p=\text{T}\) and \(q=\text{F}\).
Step 3: Antecedent True
\((p\wedge\sim q)\wedge(p\wedge r)\) is true only if \(p\), \(\sim q\) and \(r\) are all true.
With \(p=\text{T}\), \(\sim q=\text{T}\), we need \(r=\text{T}\).
Step 4: Compare the options
So \(p,q,r\) are T, F, T, which is the last option.
Final Answer:
The truth values of p, q, r are T, F, T. \[ \boxed{\text{(D) }\text{T, F, T}} \]
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