2.0: Introduction

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

So:

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

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

Notations: 

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: $\mathbb{Q} = \left\{0, \pm1, \pm\frac{1}{2}, \pm\frac{1}{3}... \right\}$
• $\mathbb{Q}$^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.