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Math Help - simple proof

  1. #1
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    simple proof

    How would you prove this, I'm stuck. Thanks

    Let A and B be sets. Then A union B = A if and only if B is subset of A.
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  2. #2
    Senior Member
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    First, you can see from the definition of a subset and from the definition of the union of two sets that for any sets A and B, A\subseteq{A\cup{B}} and B\subseteq{A\cup{B}}

    Now, to prove our "if and only if" statement, we break it up into the two direction if-then statements:

    1. If A\cup{B}=A then B\subseteq{A}.
    As noted above, B\subseteq{A\cup{B}} for any sets A and B. Thus, if A\cup{B}=A, then B\subseteq{A}.

    2. If B\subseteq{A} then A\cup{B}=A.
    Assume opposite: we have B\subseteq{A}, A\cup{B}\ne{A}. As A\subseteq{A\cup{B}}, if A\cup{B}\ne{A}, then there must be at least one element x in A\cup{B} not in A. By definition of the union of sets, every element of A\cup{B} must be an element of A or of B (or both). Thus if x\in{A\cup{B}} and x\notin{A}, then x\in{B}. Thus there must be at least one element in B not in A. But this contradicts B\subseteq{A}. Thus, we show that if B\subseteq{A} then A\cup{B}=A.

    With (1) and (2), we have proved that A\cup{B}=A if and only if B\subseteq{A}.

    -Kevin C.
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