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Advanced Calc. proof, about sets and intersection.

  1. Sep 7, 2008 #1
    1. The problem statement, all variables and given/known data
    Prove that: A is a subset of B if and only if (A intersection B)=A


    2. Relevant equations



    3. The attempt at a solutionI tried proving the right side, that is

    (A [tex]\cap[/tex] B)=A
    For two sets to be equal then they have to be subsets of each other...so:

    (A [tex]\cap[/tex] B) [tex]\subseteq[/tex] A and A [tex]\subseteq[/tex] (A [tex]\cap[/tex] B)
    So if we assume an element x [tex]\in[/tex] (A[tex]\cap[/tex]B), then by definition, x[tex]\in[/tex]A and x [tex]\in[/tex]B. Thus we proved that (A[tex]\cap[/tex]B)[tex]\subseteq[/tex]A.

    In not quite sure how to prove the opposite, because if x is an element of A, that doenst necessarily mean that x is an element of A[tex]\cap[/tex]B...so i need help with the rest of it..or if you got any other ideas on how to approach it.

    Thank you,
    Emira!
     
  2. jcsd
  3. Sep 7, 2008 #2

    HallsofIvy

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    Very good. That is exactly right!

    In not quite sure how to prove the opposite, because if x is an element of A, that doenst necessarily mean that x is an element of A[tex]\cap[/tex]B...so i need help with the rest of it..or if you got any other ideas on how to approach it.

    Thank you,
    Emira![/QUOTE]
    For the opposite, notice that you haven't used the hypothesis that A is a subset of B. If x is in A, then, because A is a subset of B it is also in B. Since it is in both A and B, it is in [itex]A\cap B[/itex]
    Now you have to prove the implication the other way: If [itex]A\cap B\subseteq A[/itex] then [itex]A\subseteq B[/itex].
     
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