Set Theory Part 8 Quiz

A ∪ B = U means B covers everything not in A (B ⊇ A′). Since A ⊂ B, B must actually equal U (because A ∪ B = U and A ⊂ B implies B = U). Thus B′ = ∅ and A′ ⊆ B′ = ∅, so A′ = B′.

(A − B)' = (A ∩ B′)' = A' ∪ B (by De Morgan's Law), since A − B = A ∩ B′.

A ∩ (B − C) = ∅ means A shares no elements with (B excluding C). This doesn't force A into B, C, or their intersection—A could avoid both regions entirely.

A ∩ B = A ∩ C with B ⊂ A only means the parts of B and C inside A coincide. C could have elements outside A, so no relationship between B and C can be guaranteed.

A − B = A ∩ B′ and A − C = A ∩ C′. If these are equal, then A ∩ B′ = A ∩ C′, so the parts of A outside B and outside C are identical, meaning the parts of A inside B and inside C are also identical: A ∩ B = A ∩ C.

The intersection of any set A with the empty set {} always yields the empty set, as no elements can be common to both. This is an identity property of sets.

Elements in A but not in (B ∪ C) must be in A, not in B, AND not in C—exactly (A − B) ∩ (A − C).

B − C = ∅ means every element of B is in C (B ⊆ C). Since A ⊂ B and B ⊂ C, transitivity gives A ⊂ C, but B ⊂ C is the direct conclusion from the given.

Union contains A and B; removing A leaves only B. A ∩ B = ∅ means A and B are disjoint. (A ∪ B) − A = B − (A ∩ B) = B − ∅ = B.

A ∪ B = A ∪ C implies (A ∪ B) − A = (A ∪ C) − A, so B − A = C − A. A ∩ B = A ∩ C implies the overlapping parts are equal. Together, these force B = C.