# Cardinality

In mathematics, the **cardinality** of a set means the number of its elements. For example, the set A = {2, 4, 6} contains 3 elements, and therefore A has a cardinality of 3. Two sets have the *same* (or *equal*) cardinality if they have the same number of elements. They have the same number of elements if and only if there is a 1-to-1 correspondence between the sets. The cardinality of the set *A* is *less than or equal to* cardinality (or *fewer than or equal* members) set *B* if and only if there is an injective function from *A* to *B*. The cardinality of the set *B* is *greater than or equal to* (or *more than or equal* members) set *B* if and only if there is an injective function from *A* to *B*.

The *cardinality* of a set is only one way of giving a number to the *size* of a set. Measure is different.

## Notation

||*A*|| is the cardinality of the set *A*.

## Finite sets

The cardinality of a finite set is a natural number. The smallest cardinality is 0. The empty set has cardinality 0. If the cardinality of the set *A* is *n*, then there is a "next larger" set with cardinality *n*+1. (For example, the set [math]A \cup \{A\}[/math]. If [math]||A|| \le ||B|| \le ||A \cup \{A\}||[/math] then either [math] ||A|| = ||B||[/math] or [math]||B|| = ||A \cup \{A\}||[/math].) There is no largest finite cardinality.

## Infinite sets

If the cardinality of a set is not finite then the cardinality is infinite.<ref>In order to simplify, we will assume the Axiom of choice</ref>

An infinite set is considered countable if they can be listed without missing any. Examples include the rational numbers, integers, and natural numbers. Such sets have a cardinality that we call [math]\aleph_0[/math] (read as: aleph null, aleph naught or aleph zero). Sets such as the real numbers are not countable. If given any finite or infinite list of real numbers, you can create a number not on that list. The real numbers have a cardinality of **c**.