Final answer:
The conditions A ∩ C = B ∩ C and A ∪ C = B ∪ C are not sufficient to conclude that sets A and B are equal. Additional information is needed to determine the equality of two sets. Mutual exclusivity and the concept of set equality must be clearly distinguished.
Step-by-step explanation:
The student's question pertains to set theory in mathematics and whether certain conditions would allow us to conclude that two sets, A and B, are equal. Specifically, we are given information about operations involving sets A, B, and C, such as their unions and intersections. For instance, if we know that A ∩ C = B ∩ C and A ∪ C = B ∪ C, we must consider additional conditions to determine if A = B.
According to the principles of set theory, if two sets have both the same intersection and union with a third set, it does not necessarily mean they are equal. They could still have elements that are different, which are not in set C. For example, consider if A = {1, 2, 3} and B = {1, 2, 4} with C being {2}. Here, A ∪ C = B ∪ C = {1, 2, 3, 4} and A ∩ C = B ∩ C = {2}, but A is not equal to B. Thus, the given conditions alone are not sufficient to conclude that A = B.
Additionally, when dealing with the probability of events, if sets A and B are mutually exclusive, which means they have no common elements, then P(A AND B) = 0. If A and C are mutually exclusive, P(A AND C) = 0. The concept of mutual exclusivity is different from the conditions of set equality.