Question

If statements $p$ and $q$ are true and $r$ and $s$ are false, then truth values of $\sim ( p \rightarrow q ) \leftrightarrow ( r \wedge s )$ and $( \sim p \rightarrow q ) \wedge ( r \leftrightarrow s )$ are respectively

• A. $F, F$
• B. $T, T$
• C. $T, F$
• D. $F, T$

Hint:

In a biconditional expression ($A \leftrightarrow B$), if both sides simplify to the exact same truth value (whether both are True or both are False), the entire expression is automatically True ($T$).

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We are given the individual truth values of four statements: $p = T$, $q = T$, $r = F$, and $s = F$. We need to evaluate the compound truth values of two distinct logical expressions.

Step 2: Key Formula or Approach:
We use the fundamental truth tables for standard logical connectives: Implication ($A \rightarrow B$): False only when $T \rightarrow F$. Conjunction ($A \wedge B$): True only when both are True. Biconditional ($A \leftrightarrow B$): True when both have identical truth values. Negation ($\sim A$): Inverts the truth value.

Step 3: Detailed Explanation:
Let's evaluate the first expression: $\sim ( p \rightarrow q ) \leftrightarrow ( r \wedge s )$ Substituting the given truth values: $$\sim ( T \rightarrow T ) \leftrightarrow ( F \wedge F )$$ $$\equiv \sim ( T ) \leftrightarrow F$$ $$\equiv F \leftrightarrow F \equiv T$$ Now, let's evaluate the second expression: $( \sim p \rightarrow q ) \wedge ( r \leftrightarrow s )$ Substituting the given truth values: $$( \sim T \rightarrow T ) \wedge ( F \leftrightarrow F )$$ $$\equiv ( F \rightarrow T ) \wedge T$$ $$\equiv T \wedge T \equiv T$$ Both compound statements evaluate to True ($T, T$).

Step 4: Final Answer:
The truth values are respectively $T, T$, matching option (B).