The language of sets and maps is basic to any presentation of mathematics. Unfortunately, in many elementary school books sets are discussed at length while maps are introduced clumsily, if at all, at a rather late stage in the story. In this chapter, by contrast, maps are introduced as early as possible. Also, by way of a change, more prominence than is usual is given to the von Neumann construction of the set of natural numbers.

Most of the material is standard. Non-standard notations include f⊢ and f⊢, to denote the forward and backward maps of subsets induced by a map f, and X! to denote the set (and in Chapter 2 the group) of permutations of a set X. The notation ω for the set of natural numbers is that used in and in. An alternative notation in common use is N.

Membership

Membership of a set is denoted by the symbol ∈, to be read as an abbreviation for ‘belongs to’ or ‘belonging to’ according to its grammatical context. The phrase ‘x is a member of X’ is denoted by x ∈ X. The phrase ‘x is not a member of X’ is denoted by x ∉ X. A member of a set is also said to be an element or a point of the set. Sets X and Y are equal, X = Y, if, and only if, each element of X is an element of Y and each element of Y is an element of X.