Review of Fundamentals

In this chapter we give a concise overview of some fundamental concepts that underlie everything we do in the rest of the book.

In section 2.1 we present the basic elements of set theory. We then gv> on to discuss the various kinds of numbers, ending with a concise treatment of the properties of real numbers. End the dimensions of economic variables. We then introduce the idea of point sets, beginning with the simplest case of intervals of the real line, and define* their most important properties from the point of view of economics: closedness, boundedness. and convexity. Next we give the general definition of a function, and set out the main properties of the types of functions most frequently encountered in economics We also define the important properties of concavity, convexity, and quasiconcavity and convexity Finally, there is a short discussion of the meaning of necessary conditions and sufficient conditions, and of how proofs are formulated, in the context of an economic example.

A set is any collection of items thought of as a whole. The collection is treated as a single object, to which mathematical operations may be applied. One way of defining a particular set is by enumeration: we simply list the items included in the set—the elements of the set. Alternatively, we can stale a specific property. If an item possesses that properly, it is an element of the set, but if it does not. it is excluded from the set. This latter method is far more generally used because defining a set by enumeration is often very cumbersome and sometimes impossible.

For example, consider the set of even numbers between I and 11. In the standard notation for describing sets, we could write

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