Introduction to Probability and Stochastic Processes with Applications

APPENDIX A

BASIC CONCEPTS ON SET THEORY

In this appendix we present some basic concepts and results from set theory that have been used throughout the book. Intuitively, a set is a well-defined grouping of objects. The objects belonging to the set are called elements. Sets are represented with uppercase Latin letters: A, B, C, M, X, … while the elements of a set are usually represented with lowercase Latin letters: a, b, c, m, x, …. To indicate that an element x belongs to the set A, we write x ∈ A, and if x is not an element of the set A, then we write x ∉ A.

Sets can be described by enumerating all of their elements or by enunciating properties that those elements must have. In the first case we say that the set is determined by extension and in the second case we say the set is determined by comprehension. Thus, we have, for example, that the set

A = {1,3,5,9}

is described by extension, while the set

B = {x : x is a rational number less than or equal to 5}

has been defined by comprehension.

A set whose elements correspond to nonnegative integers is called finite. A set that does not have any elements is called an empty set and is notated . A set is said to be infinite if it is not finite.

Two sets are said to be equal if and only if they have exactly the same elements. If all the elements of a set A are also elements of a set B we say that A is contained in B (alternatively, ...