Chapter 1
Algebraic Structures
When you are introduced to people, at first you only learn their names and
faces. Meeting them later, you begin to know them better, maybe even
become friends with them.
In the first chapter, you will be only introduced to most of the algebraic
structures considered in this book. A deeper understanding of them should
come later, through reading and problem-solving.
1.1. Introduction
If it is at all possible to define the subject of algebra precisely, then this is
the study of algebraic structures: sets on which operations are defined. By
an operation on a set M, we mean a map
M x M -• M,
i.e., a rule that assigns to every two elements of M some element of the same
set M. These elements can be numbers or objects of a different kind.
The following number sets are well-known important examples of alge-
braic structures. They have the operations of addition and multiplication:
N, the set of all natural numbers,
Z, the set of all integers,
= NU {0}, the set of all nonnegative integers,
Q, the set of all rational numbers,
E, the set of all real numbers,
, the set of all nonnegative real numbers.
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