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Mathematical Reasoning
medium
If the truth value of the Boolean expression $((\mathrm{p} \vee \mathrm{q}) \wedge(\mathrm{q} \rightarrow \mathrm{r}) \wedge(\sim \mathrm{r})) \rightarrow(\mathrm{p} \wedge \mathrm{q}) \quad$ is false then the truth values of the statements $\mathrm{p}, \mathrm{q}, \mathrm{r}$ respectively can be:
A
$\mathrm{T}\, \mathrm{F} \,\mathrm{T}$
B
$\mathrm{F\,F\,T}$
C
$\mathrm{T\,F\,F}$
D
$\mathrm{F\,T\,F}$
(JEE MAIN-2021)
Solution
| $p$ | $q$ | $r$ | $\underbrace{\mathrm{p} \vee \mathrm{q}}_{\mathrm{a}}$ | $\underbrace{\mathrm{q} \rightarrow \mathrm{r}}_{\mathrm{b}}$ | $\mathrm{a} \wedge \mathrm{b}$ | $\sim \mathrm{r}$ | $\underbrace{\mathrm{a} \wedge \mathrm{b} \wedge(\sim \mathrm{r})}_{\mathrm{c}}$ | $\underbrace{\mathrm{p} \wedge \mathrm{q}}_{\mathrm{d}}$ | $\mathrm{c} \rightarrow \mathrm{d}$ |
| $T$ | $F$ | $T$ | $T$ | $T$ | $T$ | $F$ | $F$ | $F$ | $T$ |
| $F$ | $F$ | $T$ | $F$ | $T$ | $F$ | $F$ | $F$ | $F$ | $T$ |
| $T$ | $F$ | $F$ | $T$ | $T$ | $T$ | $T$ | $T$ | $F$ | $F$ |
| $F$ | $F$ | $F$ | $T$ | $F$ | $F$ | $T$ | $F$ | $F$ | $T$ |
Standard 11
Mathematics
