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Topic: Even/Odd Cardinality, Generalized
Replies: 3   Last Post: Jan 14, 2011 2:31 PM

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Dan Luecking

Posts: 26
Registered: 11/12/08
Re: Even/Odd Cardinality, Generalized
Posted: Jan 13, 2011 9:54 PM
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On Wed, 12 Jan 2011 20:11:50 -0500 (EST), ksoileau
<> wrote:

>We say that a set X is EVEN if there exist disjoint A and B such that
>A union B equals X and A and B have the same cardinality, otherwise
>say that X is ODD. Are all infinite sets even? prove or disprove,
>assuming the Axiom of Choice. The countably infinite case is trivial,
>so assume X is uncountable.

Use Zorn's Lemma: Consider the set of all pairs (A,f)
where A is a subset of X and f is a 1-1 map from A into
X\A. Define the order, <, by (A,f) < (B,g) iff
A is a subset of B
f is the restriction of g to A.
This is a partial order and I have convinced myself
that it satisfies the hypotheses of Zorn's Lemma (for
any chain of such pairs consider the union of the sets
in each these pairs), so there is a maximal element: (M,h).

The range of h is contained in X\M, and if there are
2 elements x,y of X\M not in h(M), then add x to M to
get M' and extend h by mapping x to y to get h'. Then
(M,h) < (M',h'), contradicting the maximality of M.

Therefore X\M differs from h(M) by at most one element,
and so have the same cardinality. Since M and h(M) also
have the same cardinality, it follows that M and X\M have
the same cardinality.

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