This chapter presents a survey of all of the mathematical topics used in Chapters 3 through 15 – the mathematics needed to describe an economy with continuous supply and demand functions. Many of the topics treated here are part of the usual content of an introductory course on analysis in RN: sets, limits, convergence, open and closed sets, and continuous functions. In addition, there are topics that often are not prominent in the course on real analysis that turn out to be central to mathematical economics: convexity, separation theorems, and fixed-point theorems. This chapter assumes the student is familiar with the notation and concepts of analytic geometry. It is not a substitute for a course in real analysis (to which the student is strongly recommended).

Set theory

Let us review some basic elements of set theory.

Logical inference In mathematical logic the word implies means “leads to the logical inference that” and can be represented by the symbol of the double shaft arrow, ⇒

Definition of a setWe think of a set as a group or collection, defined by the items in the collection. A typical set might consist of all UCSD freshmen, all surfers in southern California (there is obviously some overlap here), or of the positive integers between 1 and 10. We might call a set by another name, such as a collection, a family, a class, an aggregate, or an ensemble.