Question

If the truth value of the Boolean expression $\left(\right(p\vee q)\wedge (q\to r)\wedge (\sim r\left)\right)\to (p\wedge q)$ is false, then the truth values of the statements $p,q,r$ respectively can be

• A. $FFT$
• B. $FTF$
• C. $TFF$
• D. $TFT$

Answer 1

The correct option is C $TFF$

$\therefore \left(\right(p\vee q)\wedge (q\to r)\wedge (\sim r\left)\right)\to (p\wedge q)$ is false.

$\Rightarrow (p\vee q)\wedge (q\to r)\wedge (\sim r)$ is true and $(p\wedge q)$ is false.

$\Rightarrow (p\vee q)$ is true, $(q\to r)$ is true, $(\sim r)$ is true and $(p\wedge q)$ is false.

$\Rightarrow r$ is false and exactly one out of $p$ and $q$ is false.

$\because q\to r$ is true and $r$ is false, so $q$ is also false and hence $p$ must be true.