Cardinality

The cardinality of a set is a measure of the number of elements in the set. In a finite set such as A = \{a, 3, 7, 1/2, K\} the number of elements can simply be counted; hence the cardinality of this set is |A| = 5. The cardinality of a set is usually denoted by |A| = 5. Alternatively, n(A),~\text{card}~(A), or \#A are used. This idea is generalized by the theory of cardinality and enables us to distinguish different types of infinity.
Cardinality then provides a way to systematically think about the number of elements in a set. This is also referred to as the size of a set, though this can be confusing if there is more than one possible notion of size in a given context.
There are two approaches to cardinality:
# Compare sets using bijections and injections
# Use the cardinal numbers
--> I have skipped this for now as especially the 2nd approach seemed lower priority. Should add this, possibly also a separate page on the cardinal numbers.
== Finite, countable and uncountable sets ==
Let be a set. Then, X is a:Note the connection with the natural numbers also being called the counting numbers.
* Finite set: if its cardinality is less than that of the natural numbers
*#|X| < |\mathbb{N}| = \aleph_0 
*#There exists an injective function f: X \to \mathbb{N}, but not a bijective one
* Countable set: if its cardinality is equal to that of the natural numbers
*#|X| = |\mathbb{N}| = \aleph_0 
*#There exists a bijective function f: X \to \mathbb{N}
* Uncountable set: if its cardinality is great than that of the natural numbers 
*#|X| > |\mathbb{N}| = \aleph_0 
*#There is no injective (hence no bijective) function f: X \to \mathbb{N}
== Other properties ==
* a subset of a countable set is countable
* the Cartesian product of two countable sets is countable
* the rational numbers are countable
* a countable union of countable sets is countable
* a countable sequence of closed non-empty nested intervals has a non-empty intersection
* the real numbers are not countable
The proof of the last is in math notes, and uses the second last property.
== Footnotes ==