1549901369-Elements_of_Real_Analysis__Denlinger_

1.1 The Field Properties 9

(d) Exercise 12. •

Theorem 1.1. 8 (Properties of Division and Fractions) In any fi eld F ,

(a) Vx E F , if x -/= 0, then 0 ...;-x = O;

(b) Vx E F , x -;- 1 = x; if x-/= 0 , then 1-;-x = x-^1 ;

(c) Vx E F , if x-/= 0, then (-x)-^1 = - x-^1 [that is, - (x-^1 ) ] ;

x

(d) Ify-/=0,then-=0{.;:}x =O;

y

a ac

(e) Ifb,c-/=O, th en b =be;

a c ad+ bc

(f) If b, d-/= 0, then b + d = bd ;

a c ac

(g) If b, d -/= 0, then b · d = bd;

a - a a

(h) Ifb-/=O, then-b=b= -b;

(i) If a, b-/= 0, th en (~)-

1

= ~-

b

(j) If a-/= 0, then th e equation ax + b = 0 has th e unique solution x = --.

a

Proo f. (a ) Exercise 13.
(b) Exercise 14.
(c) Exercise 15.
(d) Let y -/= 0 in F. Then y-^1 -/= 0 by Theorem 1.1.4 (f). By definition,
~ = x · y-^1. Thus,
x

• = 0 {.;:} x. y -^1 = 0
  y
  {.;:} x = O or y -^1 = 0 by Theorem 1.1.4 (e)
  {.;:} x = 0, since y -^1 -/= 0.
  (e) Suppose b, c-/= 0. Then , since c · c-^1 = 1,
  ac = (ac). (bc)-^1 by Definition 1.1.6
  be
  = (ac)(c-^1 b-^1 ) (give reasons)
  a
  = a(c · c^1 )b-^1 = a(l)b-^1 = -,;·
  (f) Exercise 16.
  (g) Exercise 17.