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

Answer: (D)

When you know that all A are B, A sits entirely inside B. If no B are C, then B has no elements in common with C. Since A is contained in B, anything that is in A also lies in B, and therefore cannot be in C. That means the statement “No A are C” must be true. The other statements don’t have to be true. A overlap with C would contradict what we just established. Saying all B are C would clash with “no B are C.” And saying all C are A would require every C to be inside A, which isn’t possible given Cs can’t be inside B where A lies; Cs could exist outside B, so that wouldn’t hold in general.

When you know that all A are B, A sits entirely inside B. If no B are C, then B has no elements in common with C. Since A is contained in B, anything that is in A also lies in B, and therefore cannot be in C. That means the statement “No A are C” must be true.

The other statements don’t have to be true. A overlap with C would contradict what we just established. Saying all B are C would clash with “no B are C.” And saying all C are A would require every C to be inside A, which isn’t possible given Cs can’t be inside B where A lies; Cs could exist outside B, so that wouldn’t hold in general.