Question

If the statement \( (p \land q) \rightarrow (r \lor \neg s) \) is False, find the truth values of \(p, q, r,\) and \(s\).

• A. \(p=T, q=T, r=F, s=T\)
• B. \(p=F, q=T, r=F, s=T\)
• C. \(p=T, q=F, r=T, s=F\)
• D. \(p=F, q=F, r=T, s=T\)

Hint:

An implication \(A \rightarrow B\) is false only in one case: when \(A\) is True and \(B\) is False. This shortcut helps solve logic questions quickly.

Answer 1

The Correct Option is A

Concept: An implication statement \[ A \rightarrow B \] is false only when \[ A = \text{True} \quad \text{and} \quad B = \text{False} \] For the given statement \[ (p \land q) \rightarrow (r \lor \neg s) \] the antecedent is \(p \land q\) and the consequent is \(r \lor \neg s\).

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

Step 2: Make the consequent False. A disjunction \(r \lor \neg s\) is false only when both parts are false. \[ r = F, \qquad \neg s = F \]

Step 3: Determine the value of \(s\). If \[ \neg s = F \] then \[ s = T \] Thus, the required truth values are \[ p = T,\quad q = T,\quad r = F,\quad s = T \]