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Thread: 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 $\displaystyle \left( {A\backslash B} \right) \equiv \left( {A \cap B^c } \right)$

    Starting with $\displaystyle B\subseteq A$ what would $\displaystyle \left( {A \cap B^c } \right)\cup A=~?$
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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 :

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

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

    $\displaystyle (p\land \lnot q) \lor q \Leftrightarrow p$ iff $\displaystyle p \lor q \Leftrightarrow p$
    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 $\displaystyle (A\cap B^c)\cup B=A$ now show that $\displaystyle B\subseteq A.$
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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 $\displaystyle (A\cap B^c)\cup B=A$ now show that $\displaystyle B\subseteq A.$
    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 $\displaystyle \left( {A\backslash B} \right) \equiv \left( {A \cap B^c } \right)$

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

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