Key Concepts and Formulas

• Truth table of logical connectives: Understanding the truth values of $\vee$ (OR), $\wedge$ (AND), $\to$ (implication), and $\sim$ (negation).
• Implication: The implication $A \to B$ is false only when $A$ is true and $B$ is false.
• Evaluating compound statements: Combining the truth values of individual statements using the truth tables of logical connectives.

Step-by-Step Solution

Step 1: Analyze the given Boolean expression.

We are given the Boolean expression: $\left( {\left( {p \vee q} \right) \wedge \left( {q \to r} \right) \wedge \left( { \sim r} \right)} \right) \to \left( {p \wedge q} \right)$ We are told that this expression is false.

Step 2: Use the truth table of implication.

For the implication $A \to B$ to be false, $A$ must be true and $B$ must be false. In our case, $A = \left( {\left( {p \vee q} \right) \wedge \left( {q \to r} \right) \wedge \left( { \sim r} \right)} \right)$ and $B = \left( {p \wedge q} \right)$. Therefore, we must have: $\left( {\left( {p \vee q} \right) \wedge \left( {q \to r} \right) \wedge \left( { \sim r} \right)} \right) = T$ and $\left( {p \wedge q} \right) = F$

Step 3: Analyze the condition $p \wedge q = F$.

The statement $p \wedge q$ is false if either $p$ is false, $q$ is false, or both are false.

Step 4: Analyze the condition $\left( {\left( {p \vee q} \right) \wedge \left( {q \to r} \right) \wedge \left( { \sim r} \right)} \right) = T$.

For the statement $\left( {\left( {p \vee q} \right) \wedge \left( {q \to r} \right) \wedge \left( { \sim r} \right)} \right)$ to be true, each of its components must be true. Therefore, we must have: $p \vee q = T$ $q \to r = T$ $\sim r = T$

Step 5: Deduce the value of $r$.

Since $\sim r = T$, we have $r = F$.

Step 6: Analyze the condition $q \to r = T$ with $r = F$.

Since $q \to r = T$ and $r = F$, $q$ must be false. If $q$ were true, then $q \to r$ would be $T \to F$, which is $F$. Therefore, $q = F$.

Step 7: Analyze the condition $p \vee q = T$ with $q = F$.

Since $p \vee q = T$ and $q = F$, $p$ must be true. If $p$ were false, then $p \vee q$ would be $F \vee F$, which is $F$. Therefore, $p = T$.

Step 8: Verify the condition $p \wedge q = F$ with $p = T$ and $q = F$.

Since $p = T$ and $q = F$, $p \wedge q = T \wedge F = F$, which satisfies the condition.

Step 9: Determine the truth values of $p, q, r$.

We have found $p = T$, $q = F$, and $r = F$. However, the provided correct answer has $r = T$, which is a contradiction. Let's re-examine the options.

If $p = T, q = F, r = T$, then:

• $p \vee q = T \vee F = T$
• $q \to r = F \to T = T$
• $\sim r = \sim T = F$
• $(p \vee q) \wedge (q \to r) \wedge (\sim r) = T \wedge T \wedge F = F$
• $p \wedge q = T \wedge F = F$
• $\left( {\left( {p \vee q} \right) \wedge \left( {q \to r} \right) \wedge \left( { \sim r} \right)} \right) \to \left( {p \wedge q} \right) = F \to F = T$

If $p = F, q = F, r = T$, then:

• $p \vee q = F \vee F = F$
• $q \to r = F \to T = T$
• $\sim r = \sim T = F$
• $(p \vee q) \wedge (q \to r) \wedge (\sim r) = F \wedge T \wedge F = F$
• $p \wedge q = F \wedge F = F$
• $\left( {\left( {p \vee q} \right) \wedge \left( {q \to r} \right) \wedge \left( { \sim r} \right)} \right) \to \left( {p \wedge q} \right) = F \to F = T$

If $p = T, q = F, r = F$, then:

• $p \vee q = T \vee F = T$
• $q \to r = F \to F = T$
• $\sim r = \sim F = T$
• $(p \vee q) \wedge (q \to r) \wedge (\sim r) = T \wedge T \wedge T = T$
• $p \wedge q = T \wedge F = F$
• $\left( {\left( {p \vee q} \right) \wedge \left( {q \to r} \right) \wedge \left( { \sim r} \right)} \right) \to \left( {p \wedge q} \right) = T \to F = F$

If $p = F, q = T, r = F$, then:

• $p \vee q = F \vee T = T$
• $q \to r = T \to F = F$
• $\sim r = \sim F = T$
• $(p \vee q) \wedge (q \to r) \wedge (\sim r) = T \wedge F \wedge T = F$
• $p \wedge q = F \wedge T = F$
• $\left( {\left( {p \vee q} \right) \wedge \left( {q \to r} \right) \wedge \left( { \sim r} \right)} \right) \to \left( {p \wedge q} \right) = F \to F = T$

Therefore, the only case where the expression is false is when $p = T, q = F, r = F$. However, the correct answer is stated to be $p = T, q = F, r = T$. There must be an error in the provided answer.

Let's re-examine the option (A) T F T:

If $p = T, q = F, r = T$, then: $(p \vee q) = T$ $(q \to r) = T$ $(\sim r) = F$ $((p \vee q) \wedge (q \to r) \wedge (\sim r)) = T \wedge T \wedge F = F$ $(p \wedge q) = F$ $F \to F = T$

This results in the expression being TRUE.

Let's re-examine the option (C) T F F:

If $p = T, q = F, r = F$, then: $(p \vee q) = T$ $(q \to r) = T$ $(\sim r) = T$ $((p \vee q) \wedge (q \to r) \wedge (\sim r)) = T \wedge T \wedge T = T$ $(p \wedge q) = F$ $T \to F = F$

This results in the expression being FALSE.

Common Mistakes & Tips

• Carefully evaluate the truth values of each connective. A single mistake can change the final result.
• Remember the truth table of implication: $A \to B$ is false only when A is true and B is false.
• Always double-check your calculations to avoid errors.

Summary

The question states that the Boolean expression is false. This occurs when the antecedent is true and the consequent is false. By analyzing the truth tables and the given conditions, we find that $p=T$, $q=F$, and $r=F$ satisfy the conditions. Therefore, option (C) is the correct one, although the provided "correct answer" was option (A).

Final Answer

The final answer is \boxed{T F F}, which corresponds to option (C).