Schaum's Outline of Discrete Mathematics, Third Edition (Schaum's Outlines)

CHAPTER 11

Properties of the

Integers

11.1Introduction

This chapter investigates some basic properties of thenatural numbers(orpositive integers), that is, the set

N={ 1 , 2 , 3 ,...}

and their “cousins,” the integers, that is, the set

Z={...,− 2 ,− 1 , 0 , 1 , 2 ,...}

(The letterZcomes from the word “Zahlen” which means numbers in German.)
The following simple rules concerning the addition and multiplication of these numbers are assumed (where
a, b, care arbitrary integers):

(a) Associative law for multiplication and addition:

(a+b)+c=a+(b+c) and (ab)c=a(bc)

(b) Commutative law for multiplication and addition:

a+b=b+a and ab=ba

(c) Distributive law:

a(b+c)=ab+ac

(d) Additive identity 0 and multiplicative identity 1:

a+ 0 = 0 +a=a and a· 1 = 1 ·a=a

(e) Additive inverse−afor any integera:

a+(−a)=(−a)+a= 0

264

Copyright © 2007, 1997, 1976 by The McGraw-Hill Companies, Inc. Click here for terms of use.