Prerequisites: Basics of Set Theory and Integers

Abstract

Chapter 1 studies some basic concepts of set theory and some properties of integers which are used throughout the book and in many other disciplines. Set theory occupies a very prominent place in modern science. There are two general approaches to set theory. The first one is called “naive set theory” initiated by Georg Cantor around 1870; the second one is called “axiomatic set theory” originated by E. Zermelo in 1908 and modified by A. Fraenkel and T. Skolem. This chapter develops naive set theory, which is a non-formalized theory using a natural language to describe sets and their basic properties. For precise description of many notions of modern algebra and also for mathematical reasoning, the concepts of relations, Zorn’s lemma, mappings (functions), cardinality of sets are very important. They form the basics of set theory and are discussed in this chapter. The set of integers plays an important role in the development of science, engineering, technology, and human civilization. In this chapter some basic concepts and properties of integers such as Peano’s axioms leading to the principle of mathematical induction, well-ordering principle, division algorithm, greatest common divisors, prime numbers, fundamental theorem of arithmetic, congruences on integers etc., are also discussed. Further studies on number theory are given in Chap. 10.