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Equivalent sets

  1. Apr 29, 2009 #1
    1. The problem statement, all variables and given/known data

    1. Suppose A-B is equivalent to B-A. show that A is equivalent to B.
    2. if A,B and C are nonempty and A cross B is equivalent to A cross C then B is equivalent to C
    Any help would be appreciated, thanks!

    3. The attempt at a solution
    I tried constructing a bijection, but that did not work out right. Any ideas?
     
    Last edited: Apr 29, 2009
  2. jcsd
  3. Apr 29, 2009 #2

    Mark44

    Staff: Mentor

    By definition, if two sets A and B are equivalent, that every element in A is also in B, and every element in B is also in A. Can you use this idea on your first problem?

    For your second problem, part of it is missing.
    Is the question "if A,B and C are nonempty and A cross B is equivalent to A cross C, then B is equivalent to C"
     
  4. Apr 29, 2009 #3
    By equivalent, I mean they have the same cardinality, not that they are equal.
     
  5. Apr 29, 2009 #4

    Mark44

    Staff: Mentor

    OK, so you know there is an bijection between A - B and B - A. It seems to me there are four cases:
    [tex]A \subset B[/tex]
    [tex]B\subset A[/tex]
    A = B
    [tex]A \cap B = \oslash[/tex]

    Can you eliminate one or more of these as possibilities, and then come up with a bijection for the remaining one(s)?

    Some examples might be helpful to get you thinking in the right way.

    1. A = {2, 4, 6, ...}, B = {1, 2, 3, ...}
    2. A = {2, 3, 4, 5, ...}, B = {1, 2, 3, 4, ...}
    3. any two sets that are equal
    4. A = {2, 4, 6, ... }, B = {1, 3, 5, ... }
     
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