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| Math Central | Quandaries & Queries |
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Question from Gloria, a teacher: how would you prove A intersect B =X iff A = X and B = X |
Hello Gloria,
Without knowing more information, we can only prove part of this statement.
If A = X and B = X, then A ∩ B = X ∩ X = X. Thus we have, A = X and B = X ⇒ A ∩ B = X. If A ∩ B = X, then we have that X ∩ A, and X ⊆ B, which is only part of what we need to solve the converse. We require more information, since in general a counterexample can be found.For instance, if A = { 1, 2, 3 } and B = { 2, 3, 4 }, then X = A ∩ B = { 2, 3 } and clearly A ≠ X, B ≠ X.
Tyler
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