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Math Help - How would you prove...

  1. #1
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    How would you prove...

    How would you prove that

    (A/B) U B = A if and only if B C A ?

    It's the iff that is giving me problems. Is there a technique for this type of proof?
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  2. #2
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    Re: How would you prove...

    Quote Originally Posted by Cairo View Post
    How would you prove that
    (A/B) U B = A if and only if B C A ?
    There is no one way to do any set theory proof.
    Here you might note that \left( {A\backslash B} \right) \equiv \left( {A \cap B^c } \right)

    Starting with B\subseteq A what would \left( {A \cap B^c } \right)\cup A=~?
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    Re: How would you prove...

    Quote Originally Posted by Cairo View Post
    How would you prove that

    (A/B) U B = A if and only if B C A ?

    It's the iff that is giving me problems. Is there a technique for this type of proof?
    This statement is equivalent to the following statement :

    (p\land \lnot q) \lor q \Leftrightarrow p iff p \lor q \Leftrightarrow p
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    Re: How would you prove...

    Quote Originally Posted by princeps View Post
    This statement is equivalent to the following statement :

    (p\land \lnot q) \lor q \Leftrightarrow p iff p \lor q \Leftrightarrow p
    Thanks, but this looks even worse to me!
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  5. #5
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    Re: How would you prove...

    Would this not be A, Plato?

    I was thinking of relabelling (B C A) as D and ((A/B) U B = A) as E and then trying to argue D implies E and also that ~D implies ~E, but not sure if this would work or even where to start.
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  6. #6
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    Re: How would you prove...

    Quote Originally Posted by Cairo View Post
    Would this not be A, Plato?
    That is correct. So you have proved it one way.

    Now suppose that (A\cap B^c)\cup B=A now show that B\subseteq A.
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  7. #7
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    Re: How would you prove...

    Quote Originally Posted by Plato View Post
    That is correct. So you have proved it one way.

    Now suppose that (A\cap B^c)\cup B=A now show that B\subseteq A.
    Using the distributive law I get

    (A U B) and (B^c U B) = .......

    But not sure how to continue this argument.

    Will have a think about it.
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    Re: How would you prove...

    Quote Originally Posted by Plato View Post
    There is no one way to do any set theory proof.
    Here you might note that \left( {A\backslash B} \right) \equiv \left( {A \cap B^c } \right)

    Starting with B\subseteq A what would \left( {A \cap B^c } \right)\cup A=~?
    Should not \left( {A \cap B^c } \right)\cup A be \left( {A \cap B^c } \right)\cup B

    And then be equal to: (B\cup A)\cap(B\cup B^c) ?
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