If the solve the problem

Question:

If $\mathrm{q}$ is false and $\mathrm{p} \wedge \mathrm{q} \leftrightarrow \mathrm{r}$ is true, then which one of the following statements is a tautology?

  1. $(\mathrm{p} \vee \mathrm{r}) \rightarrow(\mathrm{p} \wedge \mathrm{r})$

  2. $p \vee r$

  3. $p \wedge r$

  4. $(p \wedge r) \rightarrow(p \vee r)$

Correct Option: , 4

Solution:

Given $q$ is $F$ and $(p \wedge q) \leftrightarrow r$ is $T$

$\Rightarrow \mathrm{p} \wedge \mathrm{q}$ is $\mathrm{F}$ which implies that $\mathrm{r}$ is $\mathrm{F}$

$\Rightarrow \mathrm{q}$ is $\mathrm{F}$ and $\mathrm{r}$ is $\mathrm{F}$

$\Rightarrow(\mathrm{p} \wedge \mathrm{r})$ is always $F$

$\Rightarrow(\mathrm{p} \wedge \mathrm{r}) \rightarrow(\mathrm{p} \vee \mathrm{r})$ is tautology.

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