Final answer:
The equality of the intersections a ∩ b and a ∩ c does not guarantee that b = c in set theory. It indicates that b and c share the same elements that also belong to a, but each could have additional, different elements not in a.
Step-by-step explanation:
The question you have asked pertains to set theory, a fundamental concept in mathematics. Simply because a ∩ b equals a ∩ c does not necessarily mean that b = c. The intersection of sets a and b (a ∩ b) represents all elements that are both in set a and set b. If this intersection is the same as the intersection of sets a and c (a ∩ c), it suggests that sets b and c contain the same elements that also belong to set a.
However, sets b and c could still have different elements that are not in set a, meaning they can be different as long as those differences are not shared with set a. To conclusively prove that b = c, you would need to demonstrate that every element in b is also in c and vice versa, without reference to their intersection with a third set.