The logical statement

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:

  1. $(\mathrm{p} \wedge \mathrm{r}) \wedge \sim \mathrm{q}$

  2. $(\sim p \wedge \sim q) \wedge r$

  3. $\sim \mathrm{p} \vee \mathrm{r}$

  4. $(p \wedge \sim q) \vee r$


Correct Option: 1

Solution:

$\mathrm{s}[\sim(\sim \mathrm{p} \vee \mathrm{q}) \wedge(\mathrm{p} \wedge \mathrm{r})] \cap(\sim \mathrm{q} \wedge \mathrm{r})$

$\equiv[(\mathrm{p} \wedge \sim \mathrm{q}) \vee(\mathrm{p} \wedge \mathrm{r})] \wedge(\sim \mathrm{q} \wedge \mathrm{r})$

$\equiv[\mathrm{p} \wedge(\sim \mathrm{q} \vee \mathrm{r})] \wedge(\sim \mathrm{q} \wedge \mathrm{r})$

$\equiv \mathrm{p} \wedge(\sim \mathrm{q} \wedge \mathrm{r})$

$\equiv(\mathrm{p} \wedge \mathrm{r}) \sim \mathrm{q}$

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