MHT CET · Maths · Mathematical Reasoning
The statement \([(\mathrm{p} \rightarrow \mathrm{q}) \wedge \sim \mathrm{q}] \rightarrow \mathrm{r}\) is a tautology, when \(\mathrm{r}\) is equivalent to
- A \(\mathrm{p} \wedge \sim \mathrm{q}\)
- B \(q \vee p\)
- C \(\mathrm{p} \wedge \mathrm{q}\)
- D \(\sim q\)
Answer & Solution
Correct Answer
(D) \(\sim q\)
Step-by-step Solution
Detailed explanation
| p | q | r | \(p\) \( \rightarrow\) \(q\) | \(\sim q\) | \(( p \rightarrow q )\)\( \wedge \sim q\) | \([( p \rightarrow q ) \wedge\)\( \sim q ] \rightarrow r\) |
| T | T | T | T | F | F | T |
| T | T | F | T | F | F | T |
| T | F | T | F | T | F | T |
| T | F | F | F | T | F | T |
| F | T | T | T | F | F | T |
| F | T | F | T | F | F | T |
| F | F | T | T | T | T | T |
| F | F | F | T | T | T | F |
\(\therefore \quad[(p \rightarrow q) \wedge \sim q] \rightarrow r\) is a tautology when all the entries in the last column are \(T\), which is only possible when \(r \equiv \sim q\)
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