Sets and Numbers

The purpose of this chapter is to introduce two subjects that constitute the foundation of a good deal of higher mathematics, and of algebra in particular. For Chapters 2 through 5 we shall require familiarity with the elements of set theory and the real number system. This initial chapter is devoted to an exposition of the basic concepts and facts of these disciplines. Admittedly our treatment is superficial, but hopefully the reader will find it easy.

I. The Elements of Set Theory

1. The Concept of Set

The notions of set theory can be introduced in a rigorous, axiomatic way or, alternatively, in an intuitive fashion. The former method requires an excursion into logic and the foundations of mathematics. The latter enables us to get our show on the road in a quick and relatively painless way. We therefore turn at once to a heuristic description of the concept of set.

A set is a collection of objects; the nature of the objects is immaterial. The essential characteristic of a set is this: Given an object and a set, then exactly one of the following two statements is true.

(a) The given object is a member of the given set.

(b) The given object is not a member of the given set.

Examples

1. The numbers, 1 and 2, which are solutions of the equation x2 3x + 2 = 0 comprise the solution set of the given equation. We denote this set by "[1, 2}."

2. The unit circle with center at the origin of the plane is the set of points with coordinates (x, y)] satisfying the equation x2 + y2 = 1. For example, the point (1/2, [square root of 3/2]) is in the set, whereas (1, [square root of 3/2]) is not.

3. It might be tempting to speak of "the set of people who will enter the city of Chicago during 2050." But, clearly, such a collection cannot qualify as a set according to our understanding of this term (why?).

Definition 1. If an object x is a member of a set A, we say x is an element of A and write "x [member of] A." If an object y is not an element of a set B, we write "y [not member of] B."

Thus, since I is an element of the set {1, 2}, we write "1 [member of] {1, 2}"; since 3 is not an element of {1, 2}, we write "3 [not member of] {1, 2}."

Exercise: Describe a set whose elements are all sets; describe a set whose elements are sets of sets.

In Example 1, Page 1, we denoted the solution set of the equation x2 3x + 2 = 0 by "{1, 2}." This type of notation is convenient in case the elements of the set are few in number. For instance, if the set S consists of the elements a, b, c, d and no others, one writes

S = {a, b, c, d}.

In general, if a set S consists of the elements a1, a2, ..., an where n is a positive integer, then S is denoted by

(1) S = {a1, a2, ..., an}.

While the notation (1) for sets is useful as far as it goes, it will be important to have an additional notation (see Section 5, page 13).

2. Constants, Variables and Related Matters

The words "constant" and "variable" are among the most frequently used terms in mathematics. Since our usages may differ from those the reader is accustomed to, we urge that he read this section carefully.

Definition 2. A constant is a proper name, i.e., a name of a particular thing. Further, a constant names or denotes the thing of which it is a name.

Examples

1. "2" is a constant. It is the name of a particular mathematical object-a number.

2. "New York" is a constant. It is a name of one of the fifty states comprising the United States of America.

A given object may have different names, and therefore different constants may denote the same thing.

3. "1 + 1" and "8 · 1/4" are also constants, both denoting the number two.

4. New York is also known as the "Empire State." Thus "Empire State" and "New York" are names, both denoting the same geographical entity.

Variables occur in daily life as well as in mathematics. We may clarify their use by drawing upon a type of experience shared by almost all people.

Various official documents contain expressions such as (2) I, __________, do solemnly swear (or affirm) that. ... What is the purpose of the "__________" in (2)? Clearly, it is intended to hold a place in which a name, i.e., a constant, may be inserted. The variable in mathematics plays exactly the same role as does the "__________" in (2); it, too, holds a place in which constants may be inserted. However, a "__________" is clumsy for mathematical purposes. Therefore, the mathematician uses an easily written symbol, e.g., a letter of some alphabet, as a place-holder for constants. A mathematician would write (2) as, say, (3) I, x, do solemnly swear (or affirm) that ..., and the "x" is interpreted as holding a place in which a name may be inserted.

Definition 3. A variable is a symbol that holds a place for constants.

What are the constants that are permitted to replace a variable in a particular discussion? Usually an agreement is made, or understood, as to what constants are admissible as replacements. If an expression such as (2) or (3) occurs in an official document, the laws under which the document is prepared will specify the persons who may execute it. These are the individuals who may replace the variable by their names. Thus, with this variable is associated a set of persons, and the names of the persons in the set are the allowable replacements for the variable.

Definition 4. The range of a variable is the set of elements whose names are allowable replacements for the given variable.

We have said that letters are to be used as variables. It will also happen that letters will occur as constants; context will make clear whether a constant or a variable is intended.

Variables occur frequently together with certain expressions called "quantifiers." As the term implies, quantifiers deal with "how many." We shall use two quantifiers and illustrate the first as follows:

Let x be a variable whose range is the set of all real numbers. Consider the sentence

(4) For each x, if x is not zero, then its square is positive.

The meaning of (4) is

For each replacement of x by the name of a real number, if the number named is not zero, then its square is positive.

The quantifier used here is the expression "for each." Clearly, the intention is, when "for each" is used, to say something concerning each and every member of the range of the variable. For this reason we call the expression "for each" the universal quantifier.

If in place of (4) we write

(5) For each y, if y is not zero, then its square is positive, where the range of y is also the set of all real numbers, then the meanings of (4) and (5) are the same. Similarly, y can be replaced by z or some other suitably chosen 1 symbol without...