Key Concepts and Formulas

• Conditional Statement (Implication): A statement of the form "If P, then Q," denoted as $P \rightarrow Q$. It is false only when P is true and Q is false; otherwise, it is true.

• Truth Table for Implication:

  P	Q	$P \rightarrow Q$
  True	True	True
  True	False	False
  False	True	True
  False	False	True

Step-by-Step Solution

Step 1: Analyze statement (A): If 3 + 3 = 7 then 4 + 3 = 8.

• We need to determine the truth values of the antecedent (3 + 3 = 7) and the consequent (4 + 3 = 8).
• 3 + 3 = 6, which is not equal to 7. Therefore, the antecedent is false.
• 4 + 3 = 7, which is not equal to 8. Therefore, the consequent is false.
• Since the antecedent is false and the consequent is false, the implication is true. (False $\rightarrow$ False is True)

Step 2: Analyze statement (B): If 5 + 3 = 8 then earth is flat.

• We need to determine the truth values of the antecedent (5 + 3 = 8) and the consequent (earth is flat).
• 5 + 3 = 8, which is true. Therefore, the antecedent is true.
• Earth is not flat. Therefore, the consequent is false.
• Since the antecedent is true and the consequent is false, the implication is false. (True $\rightarrow$ False is False)

Step 3: Analyze statement (C): If both (A) and (B) are true then 5 + 6 = 17.

• We need to determine the truth values of the antecedent (both (A) and (B) are true) and the consequent (5 + 6 = 17).
• From Step 1, statement (A) is true. From Step 2, statement (B) is false.
• Therefore, the antecedent "both (A) and (B) are true" is false. (True AND False is False)
• 5 + 6 = 11, which is not equal to 17. Therefore, the consequent is false.
• Since the antecedent is false and the consequent is false, the implication is true. (False $\rightarrow$ False is True)

Step 4: Summarize the truth values of (A), (B), and (C).

• (A) is true.
• (B) is false.
• (C) is true.

Step 5: Compare our findings with the given options.

• Option (A): (A) is false, but (B) and (C) are true. This is incorrect because (A) is true.
• Option (B): (A) and (C) are true while (B) is false. This matches our findings.
• Option (C): (A) is true while (B) and (C) are false. This is incorrect because (C) is true.
• Option (D): (A) and (B) are false while (C) is true. This is incorrect because (A) is true.

Common Mistakes & Tips

• Understanding Implication: The most common mistake is misunderstanding the truth table for implication. Remember that $P \rightarrow Q$ is only false when P is true and Q is false.
• Careful Evaluation: Pay close attention to the truth values of the individual statements before evaluating the implications.
• AND vs. OR: Remember the difference between AND and OR. For the antecedent of (C), "both (A) and (B) are true" means (A) AND (B) must be true.

Summary

We analyzed the truth values of the three statements (A), (B), and (C) using the truth table for implication. We found that (A) is true, (B) is false, and (C) is true. This corresponds to option (B).

Final Answer

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