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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?
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2917 Upvotes
Verified Answer
The correct answer is:
$\quad(\mathrm{p} \wedge \mathrm{r}) \rightarrow(\mathrm{p} \vee \mathrm{r})$
$q$ is false and $[(p \wedge q) \leftrightarrow r]$ is true As $(p \wedge q)$ is false
$[$ False $\leftrightarrow r]$ is true
Hence $r$ is false
Option (1): says $p \vee r$,
Since $r$ is false
Hence $(p \vee r)$ can either be true or false
Option (2): says $(p \wedge r) \rightarrow(p \vee r)$
$(p \wedge r)$ is false
Since, $F \rightarrow T$ is true and
$F \rightarrow F$ is also true
Hence, it is a tautology
Option (3): $(p \vee r) \rightarrow(p \wedge r)$
i.e. $(p \vee r) \rightarrow F$
It can either be true or false
Option (4): $(p \wedge r)$
Since, $r$ is false
Hence, $(p \wedge r)$ is false.
$[$ False $\leftrightarrow r]$ is true
Hence $r$ is false
Option (1): says $p \vee r$,
Since $r$ is false
Hence $(p \vee r)$ can either be true or false
Option (2): says $(p \wedge r) \rightarrow(p \vee r)$
$(p \wedge r)$ is false
Since, $F \rightarrow T$ is true and
$F \rightarrow F$ is also true
Hence, it is a tautology
Option (3): $(p \vee r) \rightarrow(p \wedge r)$
i.e. $(p \vee r) \rightarrow F$
It can either be true or false
Option (4): $(p \wedge r)$
Since, $r$ is false
Hence, $(p \wedge r)$ is false.
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