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Let X and Y be two non-empty sets, and let A be a non-empty set such that X ∩ A = Y ∩ A = A and X ∪ A = Y ∪ A. Which of the following must be true?

1. X is a proper subset of Y
2. Y is a proper subset of X
3. X and Y are equal
4. X and Y have no common elements

Correct answer: X and Y are equal

Solution

X cap A = A means A is a subset of X, so X cup A = X. Similarly Y cup A = Y. The given X cup A = Y cup A then becomes X = Y, so the two sets must be equal.

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