Sets

8. Sets of Sets

It sometimes will happen that the object of a set are sets themselves; for example, the set of all subsets of A. In order to avoid saying “set of sets”, it is common practice to say “family of sets” or “class of sets”. Under the circumstances, and in order to avoid confusion, we sometimes will let script

letters A, B,..........

Denote families, or classes, of sets since capital letters already denote their elements.

Example 9.1: In geometry we usually say “a family of lines” or “a family of curves” since lines and curves are themselves sets of points.

Example 9.2: The set {{2,3}, {2}, {5,6}} is a family of sets. Its members are the sets {2,3}, {2} and {5,6}.

Theoretically, it is possible that a set has some members, which are sets themselves and some members which are not sets, although in any application of the theory of sets this case arises infrequently

Example 9.3: Let A = {2, {1,3}, 4, {2,5}}. Then A is not a family of sets; here some elements of A are sets and some are not.