If all A are B and no B are C, which statement must be true?

If all A are B and no B are C, which statement must be true?

• A. All C are A
• B. No A are C
• C. Some A are C
• D. All B are C

Answer: (B)

If all A are B, then A sits inside B. If no B are C, B has no elements in common with C. Since A is contained in B, nothing that is an A can be in C. So no A are C must be true. Why the others aren’t required: having all C be A would force C to lie inside A, but that would conflict with A inside B and B disjoint from C unless C is empty. Some A are C would require a shared element between A and C, which isn’t possible given the disjointness of B and C. All B are C would contradict no B are C.

If all A are B, then A sits inside B. If no B are C, B has no elements in common with C. Since A is contained in B, nothing that is an A can be in C. So no A are C must be true.

Why the others aren’t required: having all C be A would force C to lie inside A, but that would conflict with A inside B and B disjoint from C unless C is empty. Some A are C would require a shared element between A and C, which isn’t possible given the disjointness of B and C. All B are C would contradict no B are C.