Question

If the statement $(p \wedge q) \rightarrow (r \vee \neg s)$ is False (F), then the truth values of $p, q, r$ and $s$ are respectively

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

Hint:

The key rule for implication is: \[ T \rightarrow F = F \] This is the \textbf{only case} when a conditional statement becomes False. Use this rule first to determine the truth values of compound logical expressions.

Answer 1

The Correct Option is A

Concept:
For a conditional statement \(P \rightarrow Q\): \[ \text{It is False only when } P = T \text{ and } Q = F \] Also remember:

• Conjunction \(P \wedge Q\) is True only when both are True.
• Disjunction \(P \vee Q\) is False only when both are False.

Step 1: {Use the condition for implication to be false.} Given: \[ (p \wedge q) \rightarrow (r \vee \neg s) \equiv F \] An implication is False only when: \[ (p \wedge q) = T \quad \text{and} \quad (r \vee \neg s) = F \]
Step 2: {Evaluate \(p\) and \(q\).} For the conjunction: \[ (p \wedge q) = T \] both statements must be True: \[ p = T, \quad q = T \]
Step 3: {Evaluate \(r\) and \(s\).} For the disjunction: \[ (r \vee \neg s) = F \] both components must be False: \[ r = F \] \[ \neg s = F \Rightarrow s = T \]
Step 4: {Write the final truth values.} \[ (p,q,r,s) = (T,\,T,\,F,\,T) \]
Step 5: {Conclusion.} Thus, the required truth values are: \[ T,\,T,\,F,\,T \]