1549901369-Elements_of_Real_Analysis__Denlinger_

6 Chapter 1 11 The Real Number System

Proof. (a) Suppose y + x = z + x. Then, using (A3) and (A4), :Ju E F 3

x + u = 0, and (give reasons):

y = y + 0 = y + (x + u)

= (y + x) + u

= (z+x)+u

= z + (x + u)

= z + 0 = z.

Thus, y = z.

(b) Exercise l. •

Theorem 1.1.3 (Uniqueness of Identities and Inverses) In any field F,

(a) there is only one element with the property of 0 described in ( A3 );

(b) there is only one element with the property of 1 described in (M3 );

(c) Vx E F, there is only one element in F with the property of u described

in (A4);

(d) Vx =/= 0 in F, there is only one element in F with the property of u

described in (M4).

Proof. (a) Suppose 0 and O' are elements that satisfy (A3). Then

Vx E F , x + 0 = x, and Vx E F, x + O' = x, so

O' + 0 = O', and 0 + O' = 0.

Thus, O' = 0, since O' + 0 = 0 + O'.

(b) Exercise 2.

( c) Let x E F. Suppose u and v are elements of F , which satisfy the property

of u described in (A4). Then

x + u = 0, and
x +v = 0.
Thus, x + u = x + v, and so by the cancellation law [Theorem 1.1.2 (a)],
U= V.

(d) Exercise 3. •

Notation for Inverses: Since by Theorem 1.1.3, inverses are unique, we

usually denote them with special symbols. The additive inverse of an element

x E F described in (A4) is usually denoted "- x". Similarly, we usually write

1

"-" or "x-^1 " to represent the multiplicative inverse of x described in (M4).

x

Theorem 1.1.4 (Properties of Identities and Inverses) In any field F, the fol-
lowing properties hold: