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Equivalent sets

  • Thread starter guroten
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  • #1
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Homework Statement



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!

The Attempt at a Solution


I tried constructing a bijection, but that did not work out right. Any ideas?
 
Last edited:

Answers and Replies

  • #2
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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.
if A,B and C are nonempty and A cross B is equivalent to A cross then B is equivalent to C
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"
 
  • #3
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By equivalent, I mean they have the same cardinality, not that they are equal.
 
  • #4
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4,842
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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