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Proving an identity with sets

  1. Feb 6, 2014 #1
    1. The problem statement, all variables and given/known data

    e8yqcjP.png

    2. Relevant equations



    3. The attempt at a solution

    $$(A-B)\cup (C-B)=(A\cup C)-B\\ (A\cap B^{ C })\cup (C\cap B^{ C })=(A\cup C)\cap B^{ C }\\ (A\cup C)\cap B^{ C }=(A\cup C)\cap B^{ C }\\$$

    I know for algebraic proofs, proofs like these are accepted if they are reversed. But how would I reverse this proof?
     
  2. jcsd
  3. Feb 6, 2014 #2

    tiny-tim

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    hi ainster31! :smile:
    let's reverse it, as you say, and then analyse it …

    $$ (A\cup C)\cap B^{ C }=(A\cup C)\cap B^{ C }\\ (A\cap B^{ C })\cup (C\cap B^{ C })=(A\cup C)\cap B^{ C }\\(A-B)\cup (C-B)=(A\cup C)-B$$
    from the first line to the second is the distributive rule

    from the second to the third is simply applying the definition of "minus" :wink:

    (but the best way would be to start from bottom left, go up the left side, and cme down the right side (or vice versa))​
     
  4. Feb 7, 2014 #3
    That makes perfect sense. Thanks!
     
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