Question

If the statement \( (p \land q) \rightarrow (r \lor \neg s) \) is False (F), what are the truth values of \(p, q, r,\) and \(s\) respectively?

• A. \(T, T, F, T\)
• B. \(T, F, F, T\)
• C. \(T, T, T, F\)
• D. \(F, T, F, T\)

Hint:

For any logical implication \(A \rightarrow B\), the only case when the statement becomes false is when the antecedent \(A\) is true and the consequent \(B\) is false.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The question asks for the specific truth values of the propositional variables \(p, q, r,\) and \(s\) that make the entire conditional statement \( (p \land q) \rightarrow (r \lor \neg s) \) false.
Step 2: Key Formula or Approach:
A conditional statement (implication) of the form \(A \rightarrow B\) is false only in one specific case: when the antecedent (\(A\)) is true and the consequent (\(B\)) is false.
\[ A \rightarrow B \equiv \text{False} \quad \text{if and only if} \quad A \equiv \text{True} \text{ and } B \equiv \text{False} \] Step 3: Detailed Explanation:
Applying this rule to the given statement, we must have:
1. The antecedent \( (p \land q) \) must be True.
2. The consequent \( (r \lor \neg s) \) must be False.
Analyzing the antecedent:
For the conjunction \( (p \land q) \) to be True, both \(p\) and \(q\) must be True.
\[ p = \text{T}, \quad q = \text{T} \] Analyzing the consequent:
For the disjunction \( (r \lor \neg s) \) to be False, both \(r\) and \( \neg s \) must be False.
\[ r = \text{F}, \quad \neg s = \text{F} \] From \( \neg s = \text{F} \), we can determine the truth value of \(s\). If the negation of \(s\) is false, then \(s\) itself must be true.
\[ s = \text{T} \] Step 4: Final Answer:
Combining our findings, the truth values are:
\[ p = \text{T}, \quad q = \text{T}, \quad r = \text{F}, \quad s = \text{T} \] This corresponds to option (A).