Bridge to Abstract Mathematics: Mathematical Proof and Structures

302 PROPERTIES OF NUMBER SYSTEMS Chapter 9

w-(x-y)= 1-6= 1 +(-6)= 1 + 1 =2. Ontheotherhand,w-x=

w+(-x)= 1 +(-5)= 1+2=3,so(w-x)+y=3+6=2. Again,the

result predicted by a part of Theorem 5 is borne out in the example Z,,

as w - (x - y) = 2 = (w - x) + y when w = 1, x = 5, and y = 6.

A brief look at the nonfield Z, shows that Axiom 10 (the multiplicative

inverse axiom; the only field axiom that Z, fails to satisfy) must be crucial

for the proof of both Theorems 4 and 6. In Z,, 3 .2 = 0, whereas neither

3 nor 2 equals 0. Hence the conclusion of Theorem 4 fails in Z,. Of course,

there is no inconsistency, since the hypothesis of Theorem 4 also fails in Z,.

Also, in Z,, we have 2 -4 = 2. 1 (both equal 2), but 4 # 1. Cancellation of

nonzero factors, the subject of Theorem 6, does not work in the nonfield Z,.

DEFINITION 4

Let F be a field and x E F. If n is a positive integer, we define x" recursively by

x1 = x and x" = (x"-')x. If n = 0, we define x" = x0 = 1. If x#O and n is a

negative integer, we define x" = (xW1)-".

Earlier in the text (e.g., Remark 2, Article 1.5), we avoided recursive def-

inition, and would have defined xn (n E N) informally as "x times x times

x.. (n times)." By now, you should be mathematically mature enough to

prefer a precise and formal approach. Recursive definition is discussed in

detail in Article 10.1.

Properties such as xmxn = xm +", xmym = (xy)", and x - " = (xm)- l, where

m and n are positive integers and x and y are elements of an arbitrary field,

familiar from high school algebra as laws of exponents in the real number

system, are valid in the context of an arbitrary field as well.

Exercises

1. Verify the additive cancellation property of a field; that is, if x, y, z E F (F a field),
   ifx+z=y+z,thenx=y.


2. (a) Prove that the multiplicative identity (i.e., unity) 1 of a field is unique.
   *(b) Prove that the multiplicative inverse x-' of an element x in a field F is
   uniquely determined by x.


3. Prove that the zero element of a field necessarily has no multiplicative inverse.
   (Note: Axiom 10 allows only as how 0 need not have a multiplicative inverse.)


4. (a) (i) Construction addition and multiplication tables for the structure Z, of
   integers modulo 8.
   (ii) Verify five instances of the distributive law (multiplication over addi-
   tion, as in Axiom 11 of the definition of field) in Z,. How many pos-
   sible instances of distributivity are there for Z,?
   (iii) Show that Z, fails to satisfy Axiom^10 of the definition of field.
   (iv) Show that the equation (- l)a = -a is valid for each a E Z,, even
   though Z, is not a field. Is there any inconsistency between this con-
   clusion and the result of Theorem 3?