Relations and Graphs

So far we have discussed a variety of uses of graphs. Applications to everyday problems and to games and puzzles were considered. Our choice of topics had the advantage that we could deal with well-known and simple concepts. In this chapter, we shall strive to make clear that graphs are closely related to (indeed, are only a different way of formulating) some of the most fundamental concepts of mathematics in general.

A mathematical system, as we usually encounter it, consists of a set of objects or elements. For instance, we deal commonly with numbers and these may belong to more or less general types; we may discuss the set of integers, the positive numbers, the rational numbers, real numbers, imaginary numbers, or complex numbers. In algebra, we are concerned with elements which can be added, subtracted, multiplied, and so on. In geometry, we ordinarily have before us a set of points or special categories of points like straight lines, circles, planes, etc. In logic, we deal with the properties of statements of various kinds.

To construct a mathematical theory we need more than these elements; we need relations between them. Let us illustrate this: in the case of numbers, we have equal numbers a and b; in formal mathematical terminology, we write a = b. We also have numbers a and b which are different, and we write a ≠ b.