Question:
The logical statement
$[\sim(\sim \mathrm{p} \vee \mathrm{q}) \vee(\mathrm{p} \wedge \mathrm{r}) \wedge(\sim \mathrm{q} \wedge \mathrm{r})]$ is equivalent to:
Correct Option: 1
Solution:
$\mathrm{s}[\sim(\sim \mathrm{p} \vee \mathrm{q}) \wedge(\mathrm{p} \wedge \mathrm{r})] \cap(\sim \mathrm{q} \wedge \mathrm{r})$
$\equiv[(\mathrm{p} \wedge \sim \mathrm{q}) \vee(\mathrm{p} \wedge \mathrm{r})] \wedge(\sim \mathrm{q} \wedge \mathrm{r})$
$\equiv[\mathrm{p} \wedge(\sim \mathrm{q} \vee \mathrm{r})] \wedge(\sim \mathrm{q} \wedge \mathrm{r})$
$\equiv \mathrm{p} \wedge(\sim \mathrm{q} \wedge \mathrm{r})$
$\equiv(\mathrm{p} \wedge \mathrm{r}) \sim \mathrm{q}$