Q. If all Ps are Qs and some Qs are Rs, which of the following must be true?

The correct answer is (D) None of the above

Explanation:

Let’s analyze the statements and the options provided:

1. All Ps are Qs.
2. Some Qs are Rs.

Option A: All Ps are Rs

– Analysis: Since all Ps are Qs and some Qs are Rs, it doesn’t necessarily mean that all Ps are Rs. The “some Qs” that are Rs might not include any Ps.
– Conclusion: This does not have to be true.

Option B: Some Ps are Rs

– Analysis: While all Ps are Qs and some Qs are Rs, there’s no guarantee that the “some Qs” that are Rs overlap with Ps. It’s possible that none of the Ps are Rs.
– Conclusion: This does not have to be true.

Option C: No Ps are Rs

– Analysis: This is also not necessarily true. It’s possible that some Ps could be Rs if the overlapping “Qs” that are Rs include some Ps.
– Conclusion: This does not have to be true.

Option D: None of the above

– Conclusion: Since none of the options A, B, or C must necessarily be true based on the given statements, this is the correct choice.

Answer: (D) None of the above