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:
Given p = T, q = T, r = F, s = F, evaluate the truth values of two compound logical statements.

Step 2: Key Formula or Approach:
Apply standard logical operator rules: A → B is false only when T → F. A ∧ B is true only when both are true. A ↔ B is true when both have identical truth values. ∼A flips the truth value.

Step 3: Detailed Explanation:
First expression: ∼(p → q) ↔ (r ∧ s) = ∼(T → T) ↔ (F ∧ F) = ∼T ↔ F = F ↔ F = T. Second expression: (∼p → q) ∧ (r ↔ s) = (∼T → T) ∧ (F ↔ F) = (F → T) ∧ T = T ∧ T = T. Both expressions evaluate to True.

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