Question

Let $p$ and $q$ be any two logical statements and $r$ be the statement $p\to (\sim p\vee q).$ If $r$ has the truth value $\text{F},$ then the truth values of $p$ and $q$ are, respectively

• A. $\text{F, F}$
• B. $\text{T, T}$
• C. $\text{F, T}$
• D. $\text{T, F}$

Answer 1

The correct option is D $\text{T, F}$

$\begin{array}{ccccc}p& q& \sim p& \sim p\vee q& r\\ \text{T}& \text{T}& \text{F}& \text{T}& \text{T}\\ \text{F}& \text{F}& \text{T}& \text{T}& \text{T}\\ \text{T}& \text{F}& \text{F}& \text{F}& \text{F}\\ \text{F}& \text{T}& \text{T}& \text{T}& \text{T}\end{array}$

Clearly, from the above table, if $r$ has the truth value $\text{F},$ then the truth values of $p$ and $q$ are $\text{T}$ and $\text{F}$ respectively.

Alternatively, $a\to b$ is false only when the truth value of $a$ and $b$ are $\text{T}$ and $\text{F}$ respectively.
Given, $p\to (\sim p\vee q)$ is false.
$\Rightarrow p$ is true and $\sim p\vee q$ is false.
$\Rightarrow p$ is true and $q$ is false.