Chapter 0

Preliminaries

The science of Pure Mathematics, in its modern development, may claim to be the most original creation of the human spirit.

—Alfred North Whitehead

This brief chapter contains background material needed in the study of abstract algebra and introduces terms and notations used throughout the book. Presenting all this information at the beginning is preferable, because its introduction at the point it is needed interrupts the continuity of the text. Moreover, we can include enough detail here to help those readers who may be less prepared or are using the book for self-study. However, much of this material may be familiar. If so, just glance through it quickly and begin with Chapter 1, referring to this chapter only when necessary.

0.1 Proofs

The essential quality of a proof is to compel belief.

—Pierre de Fermat

Logic plays a basic role in human affairs. Scientists use logic to draw conclusions from experiments, judges use it to deduce consequences of the law, and mathematicians use it to prove theorems. Logic arises in ordinary speech with assertions such as “ if John studies hard, he will pass the course,” or “ if an integer n is divisible by 6, then n is divisible by 3.” In each case, the aim is to assert that if a certain statement is true, then another statement must also be true. In fact, if p and q denote statements, 3 most theorems take the form of an implication: “ If p is true, then q is true.” We write this in symbols as

p ⇒ q

and read it as “ ...