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Thread: Prove Set theroy

  1. #1
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    Prove Set theroy

    Prove that for any two sets A;B , we have (A n B)^c = A^c u B^c


    I think im supposed to start expansion with the de morgan laws my teacher said I was doing it wrong I'm pretty sure this is the correct approach but i dont know how to continue
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  2. #2
    Super Member Deadstar's Avatar
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    Quote Originally Posted by treetheta View Post
    Prove that for any two sets A;B , we have (A n B)^c = A^c u B^c


    I think im supposed to start expansion with the de morgan laws my teacher said I was doing it wrong I'm pretty sure this is the correct approach but i dont know how to continue
    Lol usually if your lecturer tells you it's wrong it's wrong...

    Consider an element $\displaystyle x \in (A \cap B)^c$.

    Then x is NOT contained in the intersection of A and B.

    Hence x is NOT contained in A or B.

    Hence x IS contained in $\displaystyle A^c$ and also $\displaystyle B^c$.

    So $\displaystyle x \in A^c \cup B^c$.


    Then to prove the reverse...

    Well, you try it.
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  3. #3
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    so then

    x is an element of A^c u b^c

    hence, X is an element of A^c and B^c

    so x is not contained in A or B

    so x is an element of (AnB)^c since AnB is everything thats in both A and B

    hence

    (A n B)^c = A^c u B^c

    is that the correct approach?
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  4. #4
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    Quote Originally Posted by treetheta View Post
    so then

    x is an element of A^c u b^c

    hence, X is an element of A^c and B^c No: x is an element of $\displaystyle A^c$ OR $\displaystyle B^c$. This renders the rest of the proof invalid..

    so x is not contained in A or B

    so x is an element of (AnB)^c since AnB is everything thats in both A and B

    hence

    (A n B)^c = A^c u B^c

    is that the correct approach?
    It is the correct approach.
    Now, after stating $\displaystyle x \in A^c \cup B^c \Rightarrow x \in A^c$ OR $\displaystyle x \in B^c$, separate to 2 cases:

    1) If $\displaystyle x \in A^c$, is x in $\displaystyle (A \cap B)^c$?
    2) If
    $\displaystyle x \in B^c$, is x in $\displaystyle (A \cap B)^c$?

    Try to formally write this, it usually helps..
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