Key Concepts and Formulas

• Logical Connectives: Understanding the truth tables for logical connectives like $\vee$ (OR), $\wedge$ (AND), $\sim$ (NOT), and $\to$ (IMPLICATION).
• Implication (Conditional Statement): $A \to B$ is false only when $A$ is true and $B$ is false. Otherwise, it's true.
• Truth Tables:
  • $p \vee q$ is true if at least one of $p$ or $q$ is true. It's false only when both $p$ and $q$ are false.
  • $p \wedge q$ is true only when both $p$ and $q$ are true. It's false if at least one of $p$ or $q$ is false.
  • $\sim p$ is true when $p$ is false, and false when $p$ is true.

Step-by-Step Solution

We want to find the truth values of $p, q, r$ that make the expression $\left\{ {(p \vee q) \wedge \left( {( \sim p) \vee r} \right)} \right\} \to \left( {( \sim q) \vee r} \right)$ false. This expression is of the form $A \to B$, which is false only when $A$ is true and $B$ is false. Therefore, we need to find $p, q, r$ such that $(p \vee q) \wedge ((\sim p) \vee r)$ is true and $(\sim q) \vee r$ is false.

Step 1: Analyze the condition for $(\sim q) \vee r$ to be false. $(\sim q) \vee r$ is false only when both $\sim q$ and $r$ are false. This means $q$ must be true and $r$ must be false. So, we have $q = T$ and $r = F$.

Step 2: Substitute $q = T$ and $r = F$ into the expression $(p \vee q) \wedge ((\sim p) \vee r)$ and analyze its truth value. Substituting $q = T$ and $r = F$, the expression becomes $(p \vee T) \wedge ((\sim p) \vee F)$. Since $p \vee T$ is always true regardless of the value of $p$, the expression simplifies to $T \wedge (\sim p \vee F)$. For this expression to be true, we must have $\sim p \vee F$ to be true. $\sim p \vee F$ is true only when $\sim p$ is true, which means $p$ must be false. So, we have $p = F$.

Step 3: Verify the truth values $p = F, q = T, r = F$ in the original expression. The original expression is $\left\{ {(p \vee q) \wedge \left( {( \sim p) \vee r} \right)} \right\} \to \left( {( \sim q) \vee r} \right)$. Substituting $p = F, q = T, r = F$, we have: $((F \vee T) \wedge ((\sim F) \vee F)) \to ((\sim T) \vee F)$ $(T \wedge (T \vee F)) \to (F \vee F)$ $(T \wedge T) \to F$ $T \to F$ $F$ Since the expression evaluates to false, the truth values $p = F, q = T, r = F$ make the expression false.

Step 4: Check the other options. (B) $p = T, q = T, r = F$: $((T \vee T) \wedge ((\sim T) \vee F)) \to ((\sim T) \vee F)$ $(T \wedge (F \vee F)) \to (F \vee F)$ $(T \wedge F) \to F$ $F \to F$ $T$

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

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

Only option (A) makes the expression false.

Common Mistakes & Tips

• Remember the truth table for implication. $A \to B$ is only false when $A$ is true and $B$ is false.
• When simplifying expressions, use parentheses carefully to avoid errors.
• Work systematically through each step to ensure accuracy.

Summary

To find the combination of truth values for $p, q, r$ that makes the given logical expression false, we first recognized that the expression is an implication. An implication is false only when the antecedent is true and the consequent is false. We then found the conditions for the consequent to be false, which gave us $q = T$ and $r = F$. Substituting these values into the antecedent, we found that $p$ must be false for the antecedent to be true. Therefore, the combination of truth values that makes the expression false is $p = F, q = T, r = F$. This corresponds to option (A).

Final Answer

The final answer is \boxed{p = F,q = T,r = F}, which corresponds to option (A).