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Math Help - Proof of Set Identity

  1. #1
    Junior Member mremwo's Avatar
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    Proof of Set Identity

    Show that if (A∩C)=(B∩C) then A=B, where A, B, C are sets.

    *by using a 2- way containment method without using set identities.
    *2-way method meaning showing that two sets are equal by showing they contain each other

    for example, D=E (where D=(A∩C) and E=(B∩C)) iff D⊆E ^ E⊆D.
    so i would split these two up (proof by cases) and prove:
    firstly i) D⊆E by proving (x∈D→x∈E)
    secondly ii) E⊆D by proving (x∈E→x∈D)

    *Continuing with this example, I would show how this implication (of parts i and ii) as a whole implies that A=B

    But every time I attempt to do this, I feel like I go around in circles. Or at least that I'm doing too much and making it too complicated so that I forget what I'm trying to prove. please help!
    Last edited by mremwo; October 13th 2010 at 12:44 AM. Reason: the goal is not to prove using set IDs, but by usin the defns of set, union, etc. then logical equivalences to prove the IDs
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  2. #2
    MHF Contributor Swlabr's Avatar
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    The result isn't true. For example, just take C = A \cap B.

    (An explicit counterexample would be: take A= \{1, 2, 3\}, B=\{2, 3, 4\} and C=\{2, 3\}. Then A\capC = \{2, 3\} = B\cap C...)
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  3. #3
    A Plied Mathematician
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    The result might be true if you assume that for all C,\;A\cap C=B\cap C. Might that be what the OP is getting at?
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  4. #4
    Junior Member mremwo's Avatar
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    **** WOW, sorry guys! BIG mistake
    Asking this question might make more sense if you knew that I screwed up while asking it, of course.
    What I mean to ask about is the union of these sets NOT the intersection. As in, everywhere on this problem where I put the intersection is supposed to be the union.

    I actually came up with a counterexample very similar to that of Swlabr

    -Thank you for your time
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  5. #5
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by mremwo View Post
    **** WOW, sorry guys! BIG mistake
    Asking this question might make more sense if you knew that I screwed up while asking it, of course.
    What I mean to ask about is the union of these sets NOT the intersection. As in, everywhere on this problem where I put the intersection is supposed to be the union.

    I actually came up with a counterexample very similar to that of Swlabr

    -Thank you for your time
    Yeah, this isn't true either, and your right, in that counter-examples are very similar to those for the intersection. Just take A=C, B \subsetneq C ( B a proper subset of C)...

    When working with set unions and intersections, it is always helpful to write down the venn-diagrams. Then you can verify that the result is correct, and perhaps even see a way of proving it. If the result is false, this should allow you to see this, and to come up with a counter-example!
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