Sets

A set is a basic concept in mathematics used to define collections of elements. The technical underpinnings of set theory can get a bit tricky, but for our purposes, you can think of a set as any un-ordered collection.

Notationally, we use curly braces \(\{\}\) to denote a set. For example, we could have a set consisting of three colors defined as

\[ S = \{\text{red},\;\text{green},\;\text{blue}\}. \]

Each of the three colors red, green, blue are elements of the set \(S\). The order of the elements in a set does not matter: the set above is equivalent to the set \(\{\text{blue},\;\text{green},\;\text{red}\}\). Repetition also does not matter: an element is either in the set or not.

We use the symbol \(\in\) (a funny-looking E, denoting element) to denote membership in a set:

\[ \text{red} \in S, \]

and \(\notin\) to denote that an element does not belong to a set:

\[ \text{Cleveland} \notin S. \]

We won’t do too much with sets in this text, but the basic notation is helpful to have, especially when dealing with different types of numbers.

Number systems

In digital signal processing, we use many kinds of numbers to represent different quantities. It’s helpful to have notation to specify exactly what kind of numbers we’re talking about, so here’s a brief list with their standard notations:

How natural is zero?

Note: some authors do not include 0 in the natural numbers \(\mathbb{N}\). You are likely to encounter differences of opinion when reading other sources, so beware! I’ve adopted the convention of \(0 \in \mathbb{N}\) here for many reasons, but principally because it simplifies notation overall and aligns closely to how we use these numbers when programming.

• \(\mathbb{N}\), the natural numbers: \(\{0, 1, 2, 3, \dots\}\)

• \(\mathbb{Z}\), the integers: \(\{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\}\)

• \(\mathbb{Q}\), the rational numbers (fractions): \(\left\{\frac{n}{m} ~\middle|~ n,m \in \mathbb{Z}, m \neq 0\right\}\)

• \(\mathbb{R}\), the real numbers (i.e., the continuous number line)

• \(\mathbb{C}\), the complex numbers: \(\{a + \mathrm{j}b ~|~ a,b \in \mathbb{R}\}\)

where \(\mathrm{j} = \sqrt{-1}\) is the imaginary unit.

Natural numbers (\(\mathbb{N}\)) are often used for whole number quantities, such as sample positions \(n\). Note that this means that the first sample will occur at index \(n=0\)!

Real numbers (\(\mathbb{R}\)) are often used for continuous quantities, such as angles (in radians), frequencies (in cycles/sec), or time (in seconds).

Complex numbers occupy a special place in signal processing because they turn out to be a great tool for modeling periodic processes.