Question

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

• A. $p\wedge r$
• B. $p\vee r$
• C. $(p\wedge r)\to (p\vee r)$
• D. $(p\vee r)\to (p\wedge r)$

Answer 1

The correct option is C $(p\wedge r)\to (p\vee r)$

$\because$ $q$ is $F$ and $(p\wedge q)\leftrightarrow r$ is $T$
$\Rightarrow p\wedge q$ is $F$ which implies that $r$ is $F$
$\Rightarrow q$ is $F$ and $r$ is $F$
$\Rightarrow (p\wedge r)$ is always $F$
$\Rightarrow (p\wedge r)\to (p\vee r)$ is tautology.