8 Chapter 1 • The Real Number System
Thus, x -j. 0 ::::} y = O; that is, either x = 0 or y = 0.
Part 2: For the "~" part, observe that if either x = 0 or y = 0, then xy = 0
by Part (d) and Axiom (Ml).
(f) Exercise 5.
(g) Exercise 6.
(h) Note that x+(-l)x = 1 · x+(-l)x = [1+(-1)] · x = 0 · x = x · 0 =- Thus, x + (-l)x = 0. This says that (-l)x is an additive inverse of x.
Theorem 1.1.3 ( c) says that x has only one additive inverse; namely, -x. Thus,
(-l)x = - x.
( i) Exercise 7.
(j) Exercise 8.
(k) Exercise 9. •
SUBTRACTION AND DIVISION
Results (c) and (d) of Theorem 1.1.3 make it possible to define subtraction
and division in an arbitrary field F , as follows:
Definition 1. 1.5 (Subtraction) '<Ix, y E F, define x - y = x + (-y).
Definition 1.1.6 (Division) '<Ix, y E F , if y -j. 0, define x 7 y = x · y-^1.
Note that by this definition, x 7 y = x · ( ~). We can also denote this using
"fraction" notation, :'...
yTheorem 1.1. 7 (Properties of Subtraction) In any field F,
(a) VxEF, 0-x=-x;
(b) Vx,y,zEF,x(y- z)=xy-xz, and(x-y)z=xz-yz;
(c) Vx,yEF, -(x+y)=-x-y;
(d) '<Ix, y E F, -(x - y) = y - x.
Proof. (a) Exercise 10.
(b) Let x , y, z E F. Then (give reasons, where asked)
x(y - z) = x[y + (-z)] by definition of y - z
= xy + x(-z) Why?
= xy + (-xz) Why?
= xy- xz Why?
( c) Exercise 11.