Question

If \( \{ (p \land \sim q) \land (p \land r) \} \rightarrow (p \lor q) \) has truth value false, then truth values of the statements \( p, q, r \) are respectively

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

Hint:

In logical expressions, for an implication to be false, the antecedent (left side) must be true, and the consequent (right side) must be false. Use this to simplify logical equations.

Answer 1

The Correct Option is C

Step 1: Analyzing the logical expression.
We are given the logical expression:
\[ \{ (p \land \sim q) \land (p \land r) \} \rightarrow (p \lor q) \]
and the condition that this expression has truth value false. This means that the implication \( \{ (p \land \sim q) \land (p \land r) \} \rightarrow (p \lor q) \) is false.

Step 2: Understanding the truth value of the implication.
Recall that an implication \( A \rightarrow B \) is false only when \( A \) is true and \( B \) is false. Hence, for the given implication to be false, we need:
\[ A = (p \land \sim q) \land (p \land r) \quad \text{is true and} \quad B = (p \lor q) \quad \text{is false}. \]

Step 3: Solving for \( B = (p \lor q) \).
For \( p \lor q \) to be false, both \( p \) and \( q \) must be false. Therefore, we have:
\[ p = F \quad \text{and} \quad q = F. \]

Step 4: Solving for \( A = (p \land \sim q) \land (p \land r) \).
Substitute \( p = F \) and \( q = F \) into the expression \( A \):
\[ A = (p \land \sim q) \land (p \land r) = (F \land \sim F) \land (F \land r) = (F \land T) \land (F \land r) = F. \]
Thus, the expression \( A \) is false, which does not match the required condition of \( A \) being true. Therefore, \( r \) must be true to satisfy the condition for the implication to be false.

Step 5: Conclusion.
From the above, we find that the truth values of \( p, q, r \) are:
\[ p = F, \quad q = F, \quad r = T. \]
Thus, the correct answer is option (C).
Final Answer:
The truth values of \( p, q, r \) are: \[ \boxed{F, F, T}. \]