Proving Equivalence of Sets A and B

[guroten]

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?

 

[Mark44]

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"

 

[guroten]

By equivalent, I mean they have the same cardinality, not that they are equal.

 

[Mark44]

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

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, ... }