1549901369-Elements_of_Real_Analysis__Denlinger_

1.1 The Field Properties 3

A more complete discussion of these logical symbols and others, along with

important rules for their use, are found in Appendix A. For rationale explaining

their importance in real analysis, see "Use of Logical Symbols" in the Preface.

Appendix A includes much additional material on logic and methods of proof,

while Appendix B reviews the concepts and notation of sets and functions.

1.1 The Field Properties

The real number system can be completely described in four words: it is "THE

COMPLETE ORDERED FIELD." Each of these four words deserves care-

ful examination. We shall do so in reverse order. We begin by defining the word

"field" in this section, and the remaining words in subsequent sections.

The real number system is first of all a "field." This, of course, requires a

formal definition.

Definition 1.1.1 A "field" is a set F together with two binary operations,

denoted "+ " (called addition) and "·" (called multiplication), which behave

according to the following axioms:

ADDITION AXIOMS:

(AO) \::Ix, y E F, 3 unique element x + y E F called the "sum" of x and y.

(Al) \::Ix, y E F, x + y = y + x. (commutative property of+)

(A2) \::Ix, y, z E F, x + (y + z) = (x + y) + z. (associative property of+)

(A3) ::Jelement OE F 3 \::Ix E F, x + 0 = x. (existence of a zero element)

(A4) \::Ix E F, 3u E F 3 x + u = 0. (existence of additive inverses)

MULTIPLICATION AXIOMS:

(MO) \::Ix, y E F, 3 unique element x · y E F called the "product" of x and y.

(Ml) \::/x,y E F, x · y = y · x. (commutative property of·)

(M2) \::Ix, y, z E F, x · (y · z) = (x · y) · z. (associative property of·)

(M3) ::Jelement 1EF31~0 and \::Ix E F, x · l = x. (existence of an identity

element)

(M4) \::Ix E F 3 x ~ 0, ::Ju E F 3 x · u = l. (existence of a multiplicative

inverse of every nonzero element)

DISTRIBUTIVE AXIOM:

(D) \::Ix, y , z E F, x · (y + z) = (x · y) + (x · z). (distributive property)

Comment on Multiplicative Notation: The symbol "·" is usually not

. written. As in elementary algebra, we usually write xy instead of x · y.