This appendix contains a review of some basic mathematical facts that are used throughout this book. For further details, the reader should consult a book on mathematical analysis, such as Apostal (1974), Rudin (1976), or Wade (2004).

Sets

Basic definitions. A set is a collection of objects that itself is viewed as a single entity. We write x ∈ A to indicate that x is an element of a set A; we write x ∉ A to indicate that x is not an element of A. The set that contains no elements is known as the empty set and is denoted by ø.

Let A and B denote sets. If every element of B is also an element of A we say that B is a subset of A; this is denoted by B ⊂ A. If there also exists an element of A that is not in B we say that B is a proper subset of A. If A and B have exactly the same elements we write A = B. The difference between A and B, written A \ B, is that set consisting of all elements of A that are not elements of B.

Set algebra. Let S denote a fixed set such that all sets under consideration are subsets of S and let A and B denote subsets of S. The union of A and B is the set C whose elements are either elements of A or elements of B or are elements of both; we write C = A ∪ B.