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Thread: Abstract - Prove (A-B)union(B-A)=(AunionB)-(AintersectB)

  1. #1
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    Abstract - Prove (A-B)union(B-A)=(AunionB)-(AintersectB)

    Prove:
    (A-B)union(B-A)=(AunionB)-(AintersectB).


    We need to show (A-B)union(B-A)is contained in(AunionB)-(AintersectB)
    and (AunionB)-(AintersectB)is a superset of (A-B)union(B-A).

    We begin by showing the first:
    Let x be an element of(A-B)union(B-A).
    By definition of union, x is inA-B or xis not in B-A.
    If xis in A-B, we know xis an element of A and x is not an element of B.


    This is where I've begun to get stuck. Not sure where to go next.
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  2. #2
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    I think you need to clean up your work a bit. You're trying to show that

    $\displaystyle (A-B)\cup(B-A)=(A\cup B)-(A\cap B).$

    Incidentally, both sides of this equation represent the symmetric difference of $\displaystyle A$ and $\displaystyle B$.

    What you need to show is that each side is a subset of the other. The statement you wrote

    We need to show $\displaystyle (A-B)\cup(B-A)$ is contained in $\displaystyle (A\cup B)-(A\cap B)$
    and $\displaystyle (A\cup B)-(A\cap B)$ is a superset of $\displaystyle (A-B)\cup(B-A).$
    is incorrect. The correct statement would be

    We need to show $\displaystyle (A-B)\cup(B-A)$ is contained in $\displaystyle (A\cup B)-(A\cap B)$
    and $\displaystyle (A\cup B)-(A\cap B)$ is contained in $\displaystyle (A-B)\cup(B-A).$
    For the forward direction, you start out correctly:

    Let $\displaystyle x$ be an element of $\displaystyle (A-B)\cup(B-A)$.
    But then you say this:

    By definition of union, $\displaystyle x$ is in $\displaystyle A-B$ or $\displaystyle x$ is not in $\displaystyle B-A$.
    That is incorrect. It should be this:

    By definition of union, $\displaystyle x$ is in $\displaystyle A-B$ or $\displaystyle x$ is in $\displaystyle B-A$.
    Can you continue from here, or are you still stuck?
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  3. #3
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by Ackbeet View Post
    I think you need to clean up your work a bit. You're trying to show that

    $\displaystyle (A-B)\cup(B-A)=(A\cup B)-(A\cap B).$

    Incidentally, both sides of this equation represent the symmetric difference of $\displaystyle A$ and $\displaystyle B$.

    What you need to show is that each side is a subset of the other. The statement you wrote



    is incorrect. The correct statement would be



    For the forward direction, you start out correctly:

    But then you say this:

    That is incorrect. It should be this:



    Can you continue from here, or are you still stuck?
    I always find that the easiest way to make sure you are not making mistakes with set theory stuff is to draw Venn-diagrams. They genuinely let you see what's going on.
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  4. #4
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    Thanks!!!!!!

    I am absoluteley delighted with the posts by kathrynmath and Ackbeet. I have touched on set theory and never came across a logical indication of how to attack the identities. All that is mentioned are Venn diagrams. The answers to the problems were never given. Maddening. "Each side is a subset of the other." tah dahh!

    Thanks Swlabr for your contribution, but the Venn diagrams, though obvious, can get messy and they really are not in the spirit of a mathematical proof. But you do say they are useful as a check.
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