set

Intuitively a set is a collection of entities, called its members or elements, itself considered as a single object. The fundamental principle of the theory of sets is the principle of extensionality : sets are identical if and only if they have the same members. The union of two sets is the set A ? B that has as members all the things that are members of A or B (or both). The intersection A n B is the set of things that are members of both. Sets are disjoint when they have no common members. The complement of a set B within a set A, A – B, is the set of elements that are in A but not in B. A set A is a subset of a set B when all the things that belong to A belong to B. This makes A itself a subset of A; a subset of A not itself identical with A is a proper subset of A. Sets are all members of the set-theoretic hierarchy. To obtain this we start with a list of elements: things that are not themselves sets (in case this sounds mathematically impure, we can start simply with Ø, the null set). At the bottom level we have the set of all these. At the next level we add all sets of atoms; at each level we have everything from the previous level, plus all sets of them. We then take the infinite union of all these sets, and continue ‘forever’. In fact, if we start with the null set at the lowest level, each ascending level becomes the power set of the set that constitutes the previous level.