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

Concept: In propositional logic, an implication \(A \rightarrow B\) is false only when: \[ A = \text{True} \quad \text{and} \quad B = \text{False} \] Otherwise, the implication is always true. Thus for the statement \[ (p \land q) \rightarrow (r \lor \neg s) \] to be false, we must have: \[ (p \land q) = \text{True} \quad \text{and} \quad (r \lor \neg s) = \text{False}. \]

Step 1: Analyze the antecedent \(p \land q\). For \(p \land q\) to be true, both propositions must be true: \[ p = T, \quad q = T. \]

Step 2: Analyze the consequent \(r \lor \neg s\). For the disjunction \(r \lor \neg s\) to be false, both parts must be false: \[ r = F \quad \text{and} \quad \neg s = F. \]

Step 3: Determine \(s\). Since \[ \neg s = F, \] it follows that \[ s = T. \] Thus the truth values are: \[ p = T,\quad q = T,\quad r = F,\quad s = T. \]