# Sets

## Basic definitions

A **set** is a collection of elements, such as
\[
\{1,2,3\}, \quad \{a,b,\dagger,\ddagger\}, \quad\text{ or }\quad \{\text{all yellow horses}\}.
\]
Sets are unordered. Two sets are **equal** if they contain the same elements,
\[
\{1,2,3\} = \{3,2,1\},
\]
whence the set containing no elements,
\[
\emptyset = \{\}
\]
is unique; it is called the **empty set**.

The **cardinality** of a finite set is its number of elements:
\[
|\{a,b\}| = 2 \quad\text{ and }\quad |\emptyset| = 0.
\]

**Ex.**Some well-known infinite sets are the

**natural numbers**,

^{1)}\[ \mathbb N = \{1,2,3,\ldots\},\] the

**integers**,\[\mathbb Z = \{\ldots, -1,0,1,\ldots\},\] and the

**real**, \(\mathbb R\), and

**complex numbers**, \(\mathbb C\).

^{1)}

In some textbooks also the zero element is included in the set of natural numbers.