Question:

The truth table for the statement $(\sim p \Rightarrow \sim q) \land (\sim q \Rightarrow \sim p)$ is constructed as follows: Step 1: Simplify the logical expression. The expression $(A \Rightarrow B) \land (B \Rightarrow A)$ is the definition of the biconditional $A \Leftrightarrow B$. Therefore, the given statement $(\sim p \Rightarrow \sim q) \land (\sim q \Rightarrow \sim p)$ is equivalent to $\sim p \Leftrightarrow \sim q$. The biconditional $X \Leftrightarrow Y$ is true if and only if $X$ and $Y$ have the same truth value. Thus, $\sim p \Leftrightarrow \sim q$ is true if and only if $\sim p$ and $\sim q$ have the same truth value. This condition holds precisely when $p$ and $q$ have the same truth value. Therefore, $\sim p \Leftrightarrow \sim q$ is logically equivalent to $p \Leftrightarrow q$. Step 2: Construct the truth table for $p \Leftrightarrow q$. We evaluate the truth value of $p \Leftrightarrow q$ for all possible combinations of truth values for $p$ and $q$: \begin{itemize} \item For $p=T, q=T$: $T \Leftrightarrow T$ is $T$. \item For $p=T, q=F$: $T \Leftrightarrow F$ is $F$. \item For $p=F, q=T$: $F \Leftrightarrow T$ is $F$. \item For $p=F, q=F$: $F \Leftrightarrow F$ is $T$. \end{itemize} The truth values in the last column, in the standard order corresponding to $(p,q)$ being $(T,T), (T,F), (F,T), (F,F)$, are $T, F, F, T$.