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Can you conclude that A = B if A, B, and C are sets such that: (a) A ∪ C = B ∪ C?

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Final answer:

The equality A ∪ C = B ∪ C alone is insufficient to conclude that A = B. Additional steps are needed to verify whether each element of A is in B and vice versa without the influence of set C.

Step-by-step explanation:

If A ∪ C = B ∪ C, we cannot automatically conclude that A = B. This is because the presence of set C in both unions may influence the equality.

To rigorously determine whether A equals B, we must verify that all elements of A are in B and all elements of B are in A when C is not considered. If C includes elements that are not in either A or B, or if C has common elements with either A or B that don't influence the outcome of the union, A and B might still be different. For instance, if A = {1, 2} and B = {1, 3}, with C = {2, 3}, then A ∪ C = {1, 2, 3} = B ∪ C, yet A ≠ B.

To prove A = B, we need additional information such as A ∩ B = A and A ∩ B = B, which would imply A ⊆ B and B ⊆ A, respectively, thus proving that A and B contain exactly the same elements.

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