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Re: The cardinality of the collection of all partitions of the
natural numbers equals that of the real numbers.
Posted:
Mar 17, 2011 11:16 AM
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Kerry Soileau wrote:
> Let P be the collection of all partitions of the natural numbers.
> A partition of a set X is a pairwise disjoint collection of
> sets whose union is X. I have a proof that the cardinality of P
> is the cardinality of the real numbers. Is this well-known?
> If so, would someone provide a reference?
This seems to me to be around the average difficulty level of
a problem in an undergraduate set theory or real analysis text,
so it's not something anyone would bother citing a reference
to if they wanted to use the result somewhere (and off-hand,
I don't know of a specific reference, and all my books are at
home and I'm elsewhere).
It's well-known that there is a 1-1 correspondence between the
set of all possible partitions of a set and the set of all possible
equivalence relations on a set. Since each equivalence relation
on N (the set of natural numbers) is a subset of N x N, it follows
that an upper bound on the cardinality of partitions of N is the
cardinality of the power set of N x N, which is continuum.
To show that there are at least continuum many possible partitions
of N, let E be the set of positive even integers. Then, for each
subset B of E, the set {B, N - B} is a partition of N. Moreover,
if B, B' are distinct subsets of E, then {B, N - B} is not equal
to {B', N - B'}. Thus, there are at least continuum many partitions,
since there are continuum many subsets of E.
Dave L. Renfro
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