Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications

2.2 SETS AND SAMPLE SPACES

We begin with basic definitions of sets and sample spaces, which should be a review for most readers.

Definition: Set A set is a collection of objects or numbers that represent those objects. These objects are called elements or points of the set.

Although it is possible to accurately describe a set using the original objects or outcomes in many experiments, it is often cumbersome, especially for a large number of elements. Instead, we represent the elements of a set using numbers—either integers, real numbers, or complex numbers—because it is more convenient mathematically, and is actually necessary when defining random variables in Chapter 3. For example, in experiments involving the toss of a single coin, it will be convenient to assign numbers such as and to represent the two outcomes.

Sets will be denoted by uppercase letters, usually at the beginning of the Latin alphabet, as illustrated in the following examples.

Example 2.1. Set A = {2, 4, 6, 8} consists of four integers, and set consists of all integers.

The set of all integers is represented by ...