$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

# 2.0: Introduction

• • Contributed by Pamini Thangarajah
• Professor (Mathematics & Computing) at Mount Royal University
$$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

## Sets

A set is a collection of things. These things are called elements of the set. Sets are normally denoted by using capital letters, and the elements are denoted using small letters. We write $$a \in A$$ for "a is an element of a set A", and $$a \notin A$$, for "a is not an element of a set A". $$\emptyset$$ or $$\{\}$$ denotes the empty set, which contains no element.

Example $$\PageIndex{1}$$:

Let $$A = \{1, 2, 3, 4, 5\}$$ ,

Then 1 is an element of (or belongs to) set A, we write:

$$1 \in A$$

and 0 is not an element of A, we write:

$$0 \notin A$$.

## Set Builder Notation

"Set builder notation" is used to express sets in which a pattern is present. Consider if set C is the set of all positive integers. Instead of writing down each one, how could we express set C in a general form?

This set would be written:

$$C = \{x \in \mathbb{Z} \mid 0 < x \}$$. This would read "Set C contains integers x, where x is greater than zero."

Example $$\PageIndex{2}$$:

Consider set $$D = \{1, 3, 5, 7...\}$$:

$$D$$ consists of positive, odd integers, or $$x \in \mathbb{Z}, \, x > 0, \, 2 \nmid x$$.

So:

$$D = \{x \in \mathbb{Z} \mid x > 0, \, 2 \nmid x \}$$.

Could we use any other sets to define $$x$$? Which ones would work? Which ones would not?

Example $$\PageIndex{3}$$:

Consider $$\mathbb{Q}$$, the set of rational numbers. How might we express $$\mathbb{Q}$$ in set builder notation?

Rational numbers are expressed as repeating or terminating numbers, which can be expressed as fractions: $$\frac{m}{n}$$.

In fractions, the denominator must not be zero: $$n \neq 0$$.

Also, fractions cannot have decimals as terms, so $$m$$ and $$n$$ must be $$\in \mathbb{Z}$$.

Instead of integers, if we used whole or natural numbers, we would miss out on the negative values. Thus, $$m, \, n \in \mathbb{Z}$$.

So:

$$\mathbb{Q} = \{\frac{m}{n} \mid m, \, n \in \mathbb{Z}, \, n \neq 0 \}$$

We can see, using set builder notation, that any number capable of being expressed as a fraction $$\in \mathbb{Q}$$.

Definitions:

We can use set notation to specify and help describe our standard number systems. The following standard sets are given from smallest to biggest:

• $$\mathbb{N}$$ represents the set of all natural numbers: $$\mathbb{N} = \{1, 2, 3, 4...\}$$
• $$\mathbb{W}$$ represents the set of all whole numbers: $$\mathbb{W} = \{0, 1, 2, 3...\}$$
• $$\mathbb{Z}$$ represents the set of all integers: $$\mathbb{Z} = \{ ...-2, -1, 0, 1, 2...\}$$. $$i$$ is not used because it is used for complex numbers.
• $$\mathbb{Q}$$ represents the set of all rational numbers: $$\mathbb{Q} = \{0, \pm1, \pm\frac{1}{2}, \pm\frac{1}{3}...\}$$
• $$\mathbb{Q}$$c represents the set of all irrational numbers
• $$\mathbb{R}$$ represents the set of all real numbers
• $$\mathbb{U}$$ represents the universal set, the set to which all others are a subset.