Mostly I see a partition of a set A defined as a collection of non-empty disjoint sets whose union is A.

I see one reference that allows empty sets to be included in the partition: ("Potter, M. "Set theory and its philosophy", 2004, Oxford University Press, p. 130) "Definition. A collection B of subsets of A is a partition of A if each element of A belongs to exactly one element of B.

Is there some commonly used terminology to refer to a "partition" which includes empty set(s) ????

(The context: a set of injections on the sets of a "partition" of A mapping to the set C comprises a bijection iff their images form a "partition" of C. Clearly this is true if empty sets map to empty sets, but it gets somewhat unwieldy to keep adding this to the argument.)

  • $\begingroup$ Do you mean to allow for more than one empty set? Wouldn't that also be a nonstandard definition of set? $\endgroup$ – Will Sawin Oct 20 '13 at 18:41
  • $\begingroup$ Good question. I’m looking at proofs of Cantor-Bernstein theorem. With injections f:A->B and g:B->A, it seems that in creating a bijection A is “divided” into three elements Af which is mapped to B by f; Ag by inverse of g, and Agf which can be mapped by either. Provided the images are disjoint and give B as a union the bijection is proven. However, any one or two of Af, Ag, and Agf could be empty. Can I call this division a “partition” or what else ? $\endgroup$ – Tom Collinge Oct 21 '13 at 7:20
  • $\begingroup$ Actually, if there is more than one empty element of the "partition" I don't think it would invalidate the partition as a set. The set is defined by its extension and so { {}, {}, A, B} = { {}, A, B} is a set - yes ? $\endgroup$ – Tom Collinge Oct 24 '13 at 9:05

In Bourbaki's terminology the elements of a partition may be empty (cf. E.II.4.7).

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    $\begingroup$ Thanks, but is there any term to differentiate the two possibilities. If a partition is generally understood to have non-empty elements, must one continually note condition that the some elements may be empty ? $\endgroup$ – Tom Collinge Oct 20 '13 at 12:09
  • $\begingroup$ In view of my answer "generally understood" seems to be not really appropriate. But anyway, if you use Bourbaki terminology throughout (and of course also say that you do so) then you only have to specify if you consider partitions by nonempty sets. $\endgroup$ – Fred Rohrer Oct 20 '13 at 14:21

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