Question

If the truth value of the statement $p\to (\sim q\vee r)$ is false $\left(F\right),$ then the truth values of the statements $p,q,r$ are respectively :

• A. $T,F,F$
• B. $T,F,T$
• C. $F,T,T$
• D. $T,T,F$

Answer 1

The correct option is D $T,T,F$

Given $p\to (\sim q\vee r)$ is false.
Option A) If $p=T,q=T,r=F$
so, $T\to (\sim T\vee F)T\to (F\vee F)T\to F$
which is equal to $F$

Option $B)$ If $p=T,q=F,r=T$
so, $T\to (\sim F\vee T)T\to (T\vee F)T\to T$
which is equal to $T$

Option $C)$ If $p=F,q=T,r=T$
so, $F\to (\sim T\vee T)F\to (F\vee T)T\to T$
which is equal to $T$

Option $D)$ If $p=F,q=T,r=T$
so, $F\to (\sim T\vee T)F\to (F\vee T)F\to T$
is equal to $T$

Alternate solution:
$x\to y$ is $F$ only when $x$ is $T$ and $y$ is $F.$
$\therefore$ $p\to (\sim q\vee r)$ is false only when $P$ is $T$ and $(\sim q\vee r)$ is $F$, which is possible only when both $\sim q$ and $r$ are $F$.
So, $q$ is $T$ and $r$ is $F.$