Sets and functions

Abstract

The language of sets and functions pervades mathematics, and most of the important operations in mathematics turn out to be functions or to be expressible in terms of functions. We will not define what a set is, but take as a basic (undefined) term the idea of a set X and of membership x ∈ X (x is an element of X). The negation of x ∈ X is x / ∈ X: x is not an element of X. Typically, the elements of a set will themselves be sets, underscoring the point that, in mathematics, everything is a set. A set can be described (i) as a list {x1, . . . , xn} or (ii) by giving a description of its elements, e.g. the set of positive real numbers is described via