All A are B; No B are C; Some D are A. If these statements are true, which must be true?

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Multiple Choice

All A are B; No B are C; Some D are A. If these statements are true, which must be true?

The essential idea is tracking how the sets relate to each other and what is guaranteed by those relationships. “All A are B” means A is contained in B. “No B are C” means B and C have no elements in common. “Some D are A” asserts there is at least one element that belongs to both D and A. Because A is inside B, any element of A is also in B. The element that lies in both D and A (from the third statement) is therefore also in B. But since B and C share nothing in common, that same element cannot be in C. So that element is in D and B, proving that some D are B must be true. The other options aren’t guaranteed by the given information. All A are C would conflict with A being inside B while no B are C, given that some D are A means A is nonempty. No D are A contradicts the existence of an element that is both D and A. Some D are C isn’t enforced because the overlapping portion of D with A already sits in B and cannot be C, and there’s no requirement that D must also intersect C.

The essential idea is tracking how the sets relate to each other and what is guaranteed by those relationships. “All A are B” means A is contained in B. “No B are C” means B and C have no elements in common. “Some D are A” asserts there is at least one element that belongs to both D and A.

Because A is inside B, any element of A is also in B. The element that lies in both D and A (from the third statement) is therefore also in B. But since B and C share nothing in common, that same element cannot be in C. So that element is in D and B, proving that some D are B must be true.

The other options aren’t guaranteed by the given information. All A are C would conflict with A being inside B while no B are C, given that some D are A means A is nonempty. No D are A contradicts the existence of an element that is both D and A. Some D are C isn’t enforced because the overlapping portion of D with A already sits in B and cannot be C, and there’s no requirement that D must also intersect C.

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