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More set theory

  1. Jul 29, 2006 #1
    The question is

    If A U B is a subset of A intersect B, then prove that A=B


    Now i can see this in my head and it makes sense that the elements in set A and Set B would have to be the same. The problem that i have with subset questions is how to prove that this is the case. I can start by saying

    For all x ( x is an element of (A U B) --> x is an element of (A intersect B))
    ( x is an element of (A U B) --> x is an element of A AND B))
    ( x is an element of (A U B) --> x is an element of A AND x is an element of B))

    Where do i go from here??? Any help would be great
     
  2. jcsd
  3. Jul 29, 2006 #2
    I think that this approach will work, note that the intersection of any number of sets will always be a subset of any one of those sets, and for the union of any number of sets, any set in that union will be a subset of the union. So then show that based on what you know A is a subset of B and B is a subset of A, and they are consequently equal.
     
  4. Jul 29, 2006 #3

    0rthodontist

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    This doesn't make sense--"x is an element of A AND B." What kind of set is "A AND B"? You should skip this step and go straight to saying x is an element of A and x is an element of B.

    A simple way to approach the problem is to suppose that x is an element of A. Then you want to show that it is an element of B. The reverse direction is essentially the same. Here's how your proof could start:
    "Let x be an element of A. Then x is an element of A or x is an element of B. So, x is an element of A u B."
     
  5. Jul 29, 2006 #4

    HallsofIvy

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    To prove "A= B", you have to do two things: prove "if x is an element of A, then it is an element of B" and prove "if x is an element of B, then it is an element of A".

    You might start by saying "Let x be an element of A. Then it is an element of A union B. Since A union B is a subset of A intersect B, then ..."
     
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