The Art and Craft of Problem Solving

144 CHAPTER 5 ALGEBRA

We can define "subtraction" for sets in the following natural way:

A-B:={aEA:a�B};

in other words, A - B is the set of all elements of A which are not elements of B.

Two fundamental sets are the natural numbers N : = {I, 2, 3,4, ... } and the inte­

gers Z:= {O,±I,±2, ±3, ±4, ... }.I

Usually, there is a larger "universal" set U that contains all the sets under our

consideration. This is usually understood by context. For example, if the sets that we

are looking at contain numbers, then U equals Z, JR, or C. When the universal set U is

known, we can define the complement A of the set A to be "everything" not in A; i.e.,

A: =U-A.

For example, if U = Z and A consists of all even integers, then A would consist of all

the odd integers. (Without knowledge of U, the idea of a set complement is mean­

ingless; for example, if U was unknown and A was the set of even integers, then the

elements that are "not in A" would include the odd integers, the imaginary numbers,

the inhabitants of Paris, the rings of Saturn, etc.)

A common way to define a set is with "such that" notation. For example, the set

of rational numbers Q is the set of all quotients of the form alb such that a,b E Z

and b of. O. We abbreviate "such that" by "I" or ":"; hence

Q:= {� : a,b E Z,b of. O}.

Not all numbers are rational. For example, .j2 is not rational, which we proved

on page 42. This proof can be extended, with some work, to produce many (in fact,

infinitely many) other irrational numbers. Hence there is a "larger" set of values on the

"number line" that includes Q. We call this set the real numbers JR. One can visualize

JR intuitively as the entire "continuous" set of points on the number line, while Q and

Z are respectively "grainy" and "discrete" subsets of JR.^2

Frequently we refer to intervals of real numbers. We use the notation [a , b] to

denote the closed interval {x E JR : a :s x :s b}. Likewise, the open interval (a, b) is

defined to be {x E JR : a < x < b}. The hybrids [a, b) and (a, b] are defined analogously.

Finally, we extend the real numbers by adding the new element i, defined to be the

square root of - 1; i.e., j^2 = -1. Including i among the elements of JR produces the set

of complex numbers C, defined formally by

C:= {a+bi: a,b E JR}.

The complex numbers possess the important property of algebraic closure. This

means that any finite combination of additions, subtractions, multiplications, divisions

(except by zero) and extractions of roots, when applied to a complex number, will re­

sult in another complex number. None of the smaller sets N, Z, Q or JR has algebraic

I The letter "Z" comes from the German zahlen, which means "number."

(^2) There are many rigorous ways of defining the real numbers carefully as an "extension" of the rational num­
ben,. See, for example, Chapter I in [^3 61.