Chapter 1 - SETS TO GROUPS

Abstract

This chapter discusses algebraic systems from the trivial sets to the more highly structured groups. It reviews some of the concepts that are germane to the theory of sets and that, in recent years, have made their entry even into elementary mathematics courses. The chapter an indication of how sets can be used as the building blocks for numbers. The essential relationship that exists between a set and its elements is that the latter are members of or belong to the set. There are several common ways to describe a set, the most obvious being an actual listing of its elements enclosed by braces. That is (a, b, and c) is the set of which members are the first three letters of the alphabet and (1, 2, 3, 4, 5, 6, and 7) is the set of which members are the first seven natural numbers. This method fails if one is unable to write down all the elements of a set and on this and other occasions, the set-builder notation is useful. There are many instances in everyday life and in mathematics when the elements of one set may be said to correspond to the elements of another set. If two sets A and B are so related that each element of A corresponds to a unique element of B and each element of B is the correspondent of exactly one element of A, the sets are cardinally equivalent with their elements in one-to-one correspondence. The chapter describes some concepts such as induction and well ordering, functions or mappings, semigroups, groups, and isomorphism with the help of examples.