Question

If $p \to \left( { \sim p \vee \sim q} \right)$ is false, then the truth values of $p$ and $q$ are respectively,
(A) T, F
(B) F, F
(C) F, T
(D) T, T

Answer 1

Hint: In expressions that include $ \to $ and other logical operators such as $ \sim , \vee , \wedge $; the order of operations is that $ \to $ is performed last, while $ \sim $ is performed first.

Complete step-by-step answer:
First of all, we understand the meaning of logical operators such as $ \sim , \vee , \wedge $.
Negation of a statement$\left( \sim \right)$: Let $p$ be any statement. Then, a statement that denies the statement $p$is known as negation of the statement $p$ and it is represented by $ \sim p$.
Let $p:$All triangles are equilateral.
$ \Rightarrow \sim p:$All triangles are not equilateral.
$ \wedge $ and $ \vee $: The sign $ \wedge $ is used for the word ‘And’ between two statements. For ex- $p \wedge q$ means ‘$p$ and $q$’. On the other hand, the sign $ \vee $ is used for the word ‘Or’ between two statements. For ex- $p \vee q$ means ‘$p$ or $q$’.
Given, $p \to \left( { \sim p \vee \sim q} \right)$ is false.
Now, let us make the truth table of the given statement:

$p$	$q$	$ \sim p$	$ \sim q$	$ \sim p \vee \sim q$	$p \to \left( { \sim p \vee \sim q} \right)$
T	T	F	F	F	F
T	F	F	T	T	T
F	T	T	F	T	T
F	F	T	T	T	T

From above table, if $p \to \left( { \sim p \vee \sim q} \right)$ is false then $p$ and $q$ are both true.
Hence, the answer is option (D).

Note: Another method to solve this problem is described below:
Given, $p \to \left( { \sim p \vee \sim q} \right)$ is false.
$ \Rightarrow \sim p$ is false or $ \sim q$ is false.
$ \Rightarrow p$ is true or $q$ is true.
Hence, the answer is option (D) T, T.