Key Concepts and Formulas

• Implication: The statement $p \Rightarrow q$ (p implies q) is false only when p is true and q is false.
• Disjunction: The statement $q \vee r$ (q or r) is true if either q is true, r is true, or both are true. It is false only when both q and r are false.
• Negation: The negation of a statement p, denoted by $\sim p$, has the opposite truth value of p. If p is true, $\sim p$ is false, and vice versa.

Step-by-Step Solution

Step 1: Understand the given condition. We are given that $p \Rightarrow (q \vee r)$ is false. We need to determine the truth values of p, q, and r.

Step 2: Apply the definition of implication. The implication $p \Rightarrow (q \vee r)$ is false if and only if $p$ is true and $(q \vee r)$ is false. Therefore, we have: $p = \text{True}$ $q \vee r = \text{False}$

Step 3: Determine the truth values of q and r. Since $q \vee r$ is false, it means that both q and r must be false. The disjunction (OR) is only false when both statements are false. Therefore: $q = \text{False}$ $r = \text{False}$

Step 4: Summarize the truth values. We found that $p$ is true, $q$ is false, and $r$ is false.

Common Mistakes & Tips

• Remember the truth table for implication. It's crucial to know when $p \Rightarrow q$ is false.
• Understand the difference between conjunction (AND) and disjunction (OR). $q \wedge r$ is only true if both q and r are true, while $q \vee r$ is false only if both q and r are false.
• When dealing with false implications, always start by making the left side true and the right side false.

Summary

We are given that $p \Rightarrow (q \vee r)$ is false. This occurs only when p is true and $(q \vee r)$ is false. For $(q \vee r)$ to be false, both q and r must be false. Therefore, the truth values of p, q, and r are true, false, and false, respectively.

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