Question

If $\sim p\vee q$ is T and $p\to q$ is F, then the truth value of $p\leftrightarrow q$ is?

Answer 1

We will first make a truth table for $\sim p\vee q$. Then we will make a truth table for $p\to q$. Using the given information, we will check the truth tables for possible truth values of statement $p$ and statement $q$. We will make a truth table for $p\leftrightarrow q$. Using this table, we will find the truth value of $p\leftrightarrow q$ for the truth values of statement $p$ and statement $q$ that we found.

Complete step-by-step solution
Let us make a truth table for $\sim p\vee q$.

$p$	$\sim p$	$q$	$\sim p\vee q$
T	F	T	T
T	F	F	T
F	T	T	T
F	T	F	F

The truth table for $p\to q$ is as follows,

$p$	$q$	$p\to q$
T	T	T
T	F	F
F	T	T
F	F	T

We are given that the truth value of $\sim p\vee q$ is T and that of $p\to q$ is F. From the above two truth tables, we can see that the third case in both the tables satisfies the given information. This means that when the truth value of statement $p$ is F and the truth value of statement $q$ is T, we get the truth value of $\sim p\vee q$ to be T and that of $p\to q$ to be F.
Now, let us make the truth table for $p\leftrightarrow q$.

$p$	$q$	$p\leftrightarrow q$
T	T	T
T	F	F
F	T	F
F	F	T

We have found that the value of statement $p$ is F and the truth value of statement $q$ is T. So, from the above truth table, we can conclude that the truth value of $p\leftrightarrow q$ is F.
Note: In this type of question, it is always useful to write the truth tables for given expressions. The truth values of the given expressions and the truth tables for these expressions make it easier to determine the truth values of individual statements in the expressions. These tables also prove useful in eliminating the possibility of making minor mistakes while dealing with multiple questions.