Key Concepts and Formulas

• Conditional Statement ($p \to q$): The statement "$p \to q$" (read as "if p, then q" or "p implies q") is false only when $p$ is true and $q$ is false. Otherwise, it is true.
• Negation ($\sim p$): The negation of a statement $p$ (denoted by $\sim p$) is true when $p$ is false, and false when $p$ is true.
• Disjunction ($p \vee q$): The disjunction of two statements $p$ and $q$ (denoted by $p \vee q$, read as "p or q") is true if at least one of $p$ or $q$ is true. It is false only when both $p$ and $q$ are false.

Step-by-Step Solution

Step 1: Analyze the given condition.

The given condition is that the statement $p \to (\sim p \vee \sim q)$ is false. We need to find the truth values of $p$ and $q$ that make this statement false.

Step 2: Apply the definition of conditional statement.

A conditional statement $p \to r$ is false only when $p$ is true and $r$ is false. In our case, $r$ is $(\sim p \vee \sim q)$. Therefore, for $p \to (\sim p \vee \sim q)$ to be false, we must have: $p \text{ is True and } (\sim p \vee \sim q) \text{ is False}$

Step 3: Determine the truth value of p.

From Step 2, we know that $p$ must be true. $p = T$

Step 4: Determine the truth value of $\sim p \vee \sim q$.

From Step 2, we also know that $\sim p \vee \sim q$ must be false. $\sim p \vee \sim q = F$

Step 5: Apply the definition of disjunction.

A disjunction $a \vee b$ is false only when both $a$ and $b$ are false. Therefore, for $\sim p \vee \sim q$ to be false, we must have: $\sim p = F \text{ and } \sim q = F$

Step 6: Determine the truth value of p from ~p = F.

Since $\sim p = F$, it means that $p$ must be true. This is consistent with what we found in Step 3. $p = T$

Step 7: Determine the truth value of q from ~q = F.

Since $\sim q = F$, it means that $q$ must be true. $q = T$

Step 8: Re-examine the problem statement and provided answer.

The problem statement asks for the truth values of $p$ and $q$ if $p \to (\sim p \vee \sim q)$ is false. We have determined that $p = T$ and $q = T$.

Wait! The provided answer is (A) F, F. There must be an error in my reasoning or in the provided answer. Let's re-examine the truth table.

The truth table confirms that $p \to (\sim p \vee \sim q)$ is false only when $p$ is true and $q$ is true.

However, the given answer is (A) F, F. If we plug in F, F into the expression, we get: p = F, q = F ~p = T, ~q = T ~p v ~q = T v T = T p -> (~p v ~q) = F -> T = T So, the expression is TRUE, not FALSE.

There is an error in the provided answer. The correct answer should be T, T. I believe there is a typo in the options and option (A) should be T, T instead of F, F.

Step 9: Correct the error

The given options are incorrect. The correct option should be T, T.

Common Mistakes & Tips

• Carefully consider all possible truth values when analyzing logical statements.
• When a conditional statement is false, the antecedent is true and the consequent is false.
• Always double-check your truth tables and logical deductions.

Summary

We analyzed the given conditional statement and determined the truth values of $p$ and $q$ that make the statement false. We found that $p$ and $q$ must both be true. However, the given answer is (A) F, F, which is incorrect. The correct answer should be T, T, implying an error in the provided options.

Final Answer

The final answer is \boxed{T, T}. This corresponds to a corrected option (A) which should read "T, T".