Step 1: Understanding the Question:
The question asks us to determine the truth value (True or False) of three conditional statements (implications) labeled (A), (B), and (C), and then to identify which of the final options correctly describes these truth values.
Step 2: Key Formula or Approach:
A conditional statement "If P, then Q" (written as P \( \rightarrow \) Q) is only false when the hypothesis P is true and the conclusion Q is false. In all other cases, the implication is true.
Truth Table for P \( \rightarrow \) Q:
- T \( \rightarrow \) T is True
- T \( \rightarrow \) F is False
- F \( \rightarrow \) T is True
- F \( \rightarrow \) F is True
Step 3: Detailed Explanation:
Let's analyze each statement's truth value.
Statement (A): If 4+3 = 8, then 5+3=9
- Hypothesis (P): "4+3 = 8" is False.
- Conclusion (Q): "5+3 = 9" is False.
- The implication is F \( \rightarrow \) F, which is True. So, statement (A) is True.
Statement (B): If 6 + 4 = 10, then moon is flat
- Hypothesis (P): "6 + 4 = 10" is True.
- Conclusion (Q): "moon is flat" is False.
- The implication is T \( \rightarrow \) F, which is False. So, statement (B) is False.
Statement (C): If both (A) and (B) are true, then 5 + 6 = 17
- Hypothesis (P): "both (A) and (B) are true". We found that (A) is True and (B) is False. Therefore, the statement "both (A) and (B) are true" is False.
- Conclusion (Q): "5 + 6 = 17" is False.
- The implication is F \( \rightarrow \) F, which is True. So, statement (C) is True.
Summary of Truth Values:
- Statement (A) is True.
- Statement (B) is False.
- Statement (C) is True.
Evaluate the Options:
- Option (A): (A) is true while (B) and (C) are false. (Incorrect, C is true).
- Option (B): (A) and (B) are false, while (C) is true. (Incorrect, A is true).
- Option (C): (A) and (C) are true, while (B) is false. (Correct).
- Option (D): (A) is false, but (B) and (C) are true. (Incorrect, A is true and B is false).
Step 4: Final Answer:
The correct description of the truth values is given in option (C).