A Axiom of choice: Replacing a strong condition with a weaker condition

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The discussion centers on the implications of a set-theory theorem regarding the Axiom of Choice (ACh). It explores whether a weaker condition, A∩C≈B∩D, can replace a stronger one, A∩C=∅ & B∩D=∅, while still maintaining the validity of the theorem. The corrected implication suggests that if A≈B, C≈D, and A∩C≈B∩D, then A∪C≈B∪D. Some participants argue that both implications could potentially be proven without invoking ACh, indicating they may be independent of it. The conversation highlights the complexity of the relationships between these conditions and the Axiom of Choice.
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This set-theory theorem is very easy to prove:
(*) if A≈B & C≈D & A∩C=∅ & B∩D=∅ then A∪C≈B∪D
It seems intuitive that if one replaces the strong
A∩C=∅ & B∩D=∅
condition by the weaker
A∩C≈B∩D
the implication
(**) if A≈B & C≈D & A∩C≈B∪∩D then A∪C≈BD
still holds.
(**) does not seem to be much stronger than (*), nevertheless I have been able to prove (**) only using Ax of Choice (ACh). This suggested to me that (**) might be other equivalent to ACh, but I have not found it in the standard lists, nor I have been able to prove that (**) implies ACh.
Does anybody have any clue on this?
 
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Sorry there was a typo in (**)

Corrected:
(**) if A≈B & C≈D & A∩C≈B∩D then A∪C≈B∪D
 
What do the squiggly equal signs stand for? Equal cardinality?
 
yes
 
jose diez said:
Does anybody have any clue on this?
I haven't worked through it, but isn't it possible to prove both (*) or (**) by induction without invoking AC? In this case, they would be independent of AC so cannot be equivalent.
 
jose diez said:
Corrected:
(**) if A≈B & C≈D & A∩C≈B∩D then A∪C≈B∪D
Shooting from hip, I'd say choice is not necessary for this to occur. To prove choice, we would have an arbitrary family of nonempty sets and we must find a choice function. The condition (**) is formulated in only finite terms. I don't see how it provides an angle to tackle with infinite families.

That said, I don't have a counterexample.
 
If you subtract ##A\cap C## from the left side of everything, and ##B\cap D## on the right side, then you have reduced to the case of no intersection as long as you can show the following result:

If ##B\subset A##, ##D\subset C##, ##D\approx B##, ##A\approx C##, then ##A-B \approx C- D##
 
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