![]()
Chapter 1
SETS
11. Introduction
In this chapter we present the essential terminology, notation, and facts pertaining to sets and members of sets, the most elementary ingredients of current mathematical discourse. Successive chapters will introduce other fundamental ingredients, such as mappings, relations, numbers . . .
We do not aim either at a philosophical elucidation of these concepts, or at a rigorous account of the foundations of mathematics as they are currently understood. These are specialized subjects, attractive in their own right, but of little immediate concern to most practicing mathematicians or users of mathematics; to engage seriously in their study requires considerable mathematical experience and maturity. We merely intend to clarify the usage of the fundamental concepts, derive their simplest properties and relationships, and make the language they constitute available for use.
Underlying all mathematical discourse are concepts and rules of logic. For an exposition of the relevant logical tools that is particularly well suited to our approach, we refer to Chapter 2 of A. M. Gleason, Fundamentals of Abstract Analysis. The book as a whole is recommended for its choice of contents and its professional style. Although the usage adopted in it differs in many particulars from ours, the book is an excellent aid to understanding. Chapter 1 and part of Chapter 3 of that book should also be studied in conjunction with the present chapter.
Our use of equality is exclusively as follows: the assertion a = b means that the object (designated by the symbol) a and the object (designated by the symbol) b are one and the same. In practice, what stands on either side of = may be a complicated array of typographical symbols. The negation of the assertion a = b is denoted by a ≠ b, and if it holds we say that a and b are distinct objects, or that a is distinct from b.
The symbols := and =: are used in definitions: a := b or, equivalently, b =: a means that a is defined to be (equal to) b. The colon stands on the side of the definiendum a, the term to be defined, in order to contrast it with the definiens b, the defining term. When the definition of an object is given in words, we distinguish the words constituting the definiendum by means of boldface type. For example, “A square is defined to be a rectangle with equal sides,” or “A rectangle with equal sides is called a square.”
Similar usages occur in the definition of a property of an object by means of an appropriate predicate. Thus, “A number n is said to be even if 2 divides n” (in this style of definition it would be redundant to add “and only if”). In slightly more symbolic form we might write
For every number n, (n is even) :⇔ (2 divides n);
the symbol :⇔ may be read “means by definition that” or “is equivalent by definition to”.
12. Sets and their members
In our presentation we give no formal definition of the concept of set. Speaking vaguely, whenever objects are thought of as collected into a definite whole, a set describes this state of affairs. A set is determined...
