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.