Question

If the truth value of the statement $(P\wedge (\sim R\left)\right)\to \left(\right(\sim R)\wedge Q)\text{is}F$, then the truth value of which of the following is $F$?

Answer 1

$\underset{X}{\underset{}{P\wedge (\sim R)}}\to \underset{Y}{\underset{}{\left(\right(\sim R)\wedge Q)}}=\text{False}$
$X\to Y=\text{False}$
$\begin{array}{ccc}\text{X}& \text{Y}& \text{X}\to \text{Y}\\ \text{F}& \text{F}& \text{T}\\ \text{T}& \text{T}& \text{T}\\ \text{F}& \text{T}& \text{T}\\ \text{T}& \text{F}& \text{F}\end{array}$
$P\wedge \sim R=T\text{and}(\sim R)\wedge Q=F$
$\Rightarrow P=T$
$\sim R=T\Rightarrow R=F$
$\Rightarrow P=T,Q=F\text{and}R=F$
$T\wedge Q=F$
$\Rightarrow Q=F$
Now $\sim (R\vee Q)\to \sim P$
$\sim (F\vee F)\to F$
$F\to F=\text{False}$