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Math Help - Set theory proof help

  1. #1
    primitive
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    Set theory proof help

    I need to prove that for all sets A and B, if the complement of A is a subset of B then the union of A and B is equal to U, the universal set.

    So far, I've come up with this:

    Suppose the complement of A is a subset of B and the union of A and B is not equal to U. Let x be part of the union of A and B. By definition of union, x is a part of A or x is a part of B. Here is where I get stuck...what should I do next? Am I on the right track?

    Thanks for any help.
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  2. #2
    MHF Contributor

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    Given that A^c  \subseteq B.
    It is always true that A \cup B \subseteq U.
    So suppose that x \in U if x \in A then we are done.
    If x \notin A the by the given x \in B and we are done.
    Therefore A \cup B = U
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