ABSTRACT
The objective of Chapter 2 is to provide relevant background in elementary real analysis essential for understanding later chapters. It is emphasized that set theory provides a method for developing abstract models of concepts. This is important not only for models of intelligent systems but also for explicitly formulating concepts in mathematics. A variety of important sets are introduced which are required for understanding the discussion in later chapters. These sets include: the set of natural numbers, the set of real numbers, the extended real numbers, countable sets, and the partition of a set. The chapter also emphasizes that a relation between two sets corresponds to a semantic network and may be represented as a graph. A number of important relations and graphs are introduced including: reflexive relations, symmetric relations, transitive relations, complete relations, directed graphs, directed acyclic graphs, undirected graphs, functions, function restrictions, function extensions, and bounded functions. In addition, selected topics about metric spaces are included such as: open sets, closed sets, bounded sets, hyperrectangles, the distance from a point to a set, the neighborhood of a set, and convex sets.
