family of subsets

Given any set SS, a family of subsets of SS consists of an index set II and and a function from II to the power set of SS. This is actually nothing special; this is the normal meaning of ‘family’ when taking a family of things that form a set: a family of foos is any function to the set of foos.

However, in predicative mathematics, we do not accept that the subsets of SS form a set; that is, we believe that the power set 𝒫S\mathcal{P}S is really a proper class. So on the face of it, a family of subsets would also be a large thing. However, there is an alternative definition that is not only appropriate predicatively but also often provides a useful perspective in any case: A family of subsets of SS consists of an index set II and a binary relation \in between elements of SS and elements of II.

Note that we may have xAxBx \in A \Longleftrightarrow x \in B for all xx without having A=BA = B; this is because (as in any family) two different indices may index the same subset. However, many of the operations performed on families of subsets, such as intersection and union, depend only on the image of the function I𝒫SI \to \mathcal{P}S. So it is common to assume that a family of subsets of SS is simply a subset of 𝒫S\mathcal{P}S. (Besides this, not all authors agree with this page on even the general meaning of the word ‘family’.)

Last revised on August 19, 2010 at 16:39:15. See the history of this page for a list of all contributions to it.