Principles of Mathematics in Operations Research

124 9 Number Systems

Proposition 9.2.4 In a field F, the following properties hold:

(a) x + y = x + z=>y = z (cancelation law for addition).

(b) x + y = x =>• y — 0.

(c) x + y — 0=^y= —x.

(d) —{—x) = x.

(e) x y£ 0 and xy = xz => y = z (cancelation law for multiplication).

(f) x ^ 0 and xy = x => y — 1.

(g) x ^ 0 and xy = 1 =>- y = £.

(h) x^O, j£fc = x.

(i) Vx e F, Ox = 0.

(j) x ^ 0 and 2/^0, £/ien xy ^ 0 (no zero divisors).

(k) Wx,ye F, (-x){-y) = xy.

Definition 9.2.5 Let F be an ordered set and a field of F is an ordered field

if

i) x, y, z £ F and x~<y=>x + z-<y + z,

ii) xyO, y y 0 =>• xy y 0.

If xy 0, call x as positive, If x -< 0, call x as negative.

Example 9.2.6 S = Q is an ordered field.

Proposition 9.2.7 Let F be an ordered field. Then,

(a) x y 0 «• -x -< 0.

(b) x y 0 and y < z => xy -< xz.

(c) x -< 0 and y < z => xyy xz.

(d) x =^ 0 => x^2 y 0. /n particular 1 >- 0.

fej(Mx^y^0^±^±.

Proof. F is an ordered field.

(a) Assume x >>- 0 =>• x + (-x) y 0 + (-x) => 0 > x.

-x-<0=>-x + x-<0 + x=>0^:x.

(b) Let x >- 0 and y < z => 0 -< z - y => 0 -< x(z - y) = xz — xy =$ xy < xz.
(c) a; -< 0 and y < z =>• -x >- 0 and z - y >- 0 =>• —x(z — y)y0=> x(z — y)<
0 => xz -< xy.
(d) x =£ 0 => x y 0 =$> (y = x in (b)) x^2 >- 0 or
x-<0=»-x^0(j/ = -x) => (-x)(-x) = x^2 y 0.
(e) Let x >- 0. Show \ y 0. If not, ± ^ 0 =* (x >- 0), x± = 1 < 0,
Contradiction!
Assume 0~<x-<y=>±y0, ^ y 0, therefore (by (b))

i±>-ol x y 1

x -< y J

1 1

> => < -.

y x

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