# Cardinality

For other uses, see Cardinality (disambiguation).

## Comparing sets

While the cardinality of a finite set is just the number of its elements, extending the notion to infinite sets usually starts with defining the notion of comparison of arbitrary sets (some of which are possibly infinite).

### Definition 1: |A| = |B|

- Two sets A and B have the same cardinality if there exists a bijection (a.k.a., one-to-one correspondence) from A to B, that is, a function from A to B that is both injective and surjective. Such sets are said to be equipotent, equipollent, or equinumerous. This relationship can also be denoted A ≈ B or A ~ B.

- For example, the set E = {0, 2, 4, 6, ...} of non-negative even numbers has the same cardinality as the set N = {0, 1, 2, 3, ...} of natural numbers, since the function f(n) = 2n is a bijection from N to E (see picture).

### Definition 2: |A| ≤ |B|

- A has cardinality less than or equal to the cardinality of B, if there exists an injective function from A into B.

### Definition 3: |A| < |B|

- A has cardinality strictly less than the cardinality of B, if there is an injective function, but no bijective function, from A to B.

- For example, the set N of all natural numbers has cardinality strictly less than its power set P(N), because g(n) = { n } is an injective function from N to P(N), and it can be shown that no function from N to P(N) can be bijective (see picture). By a similar argument, N has cardinality strictly less than the cardinality of the set R of all real numbers. For proofs, see Cantor's diagonal argument or Cantor's first uncountability proof.

If |A| ≤ |B| and |B| ≤ |A|, then |A| = |B| (a fact known as Schröder–Bernstein theorem).

The axiom of choice is equivalent to the statement that |A| ≤ |B| or |B| ≤ |A| for every A, B.

## Cardinal numbers

Main article: Cardinal number

In the above section, "cardinality" of a set was defined functionally.

In other words, it was not defined as a specific object itself.

However, such an object can be defined as follows.

The relation of having the same cardinality is called equinumerosity, and this is an equivalence relation on the class of all sets.

The equivalence class of a set A under this relation, then, consists of all those sets which have the same cardinality as A.

There are two ways to define the "cardinality of a set":

- The cardinality of a set A is defined as its equivalence class under equinumerosity.
- A representative set is designated for each equivalence class. The most common choice is the initial ordinal in that class. This is usually taken as the definition of cardinal number in axiomatic set theory.

Assuming the axiom of choice, the cardinalities of the infinite sets are denoted

## Finite, countable and uncountable sets

If the axiom of choice holds, the law of trichotomy holds for cardinality.

Thus we can make the following definitions:

## Infinite sets

### Cardinality of the continuum

Main article: Cardinality of the continuum

The continuum hypothesis states that there is no cardinal number between the cardinality of the reals and the cardinality of the natural numbers, that is,

However, this hypothesis can neither be proved nor disproved within the widely accepted ZFC axiomatic set theory, if ZFC is consistent.

Cardinal arithmetic can be used to show not only that the number of points in a real number line is equal to the number of points in any segment of that line, but that this is equal to the number of points on a plane and, indeed, in any finite-dimensional space.

These results are highly counterintuitive, because they imply that there exist proper subsets and proper supersets of an infinite set S that have the same size as S, although S contains elements that do not belong to its subsets, and the supersets of S contain elements that are not included in it.

The first of these results is apparent by considering, for instance, the tangent function, which provides a one-to-one correspondence between the interval (−½π, ½π) and R (see also Hilbert's paradox of the Grand Hotel).

The second result was first demonstrated by Cantor in 1878, but it became more apparent in 1890, when Giuseppe Peano introduced the space-filling curves, curved lines that twist and turn enough to fill the whole of any square, or cube, or hypercube, or finite-dimensional space.

These curves are not a direct proof that a line has the same number of points as a finite-dimensional space, but they can be used to obtain such a proof.

- the set of all subsets of R, i.e., the power set of R, written P(R) or 2
- the set R of all functions from R to R

Both have cardinality

## Examples and properties

## Union and intersection

If A and B are disjoint sets, then

From this, one can show that in general, the cardinalities of unions and intersections are related by the following equation:

## See also

Credits to the contents of this page go to the authors of the corresponding Wikipedia page: en.wikipedia.org/wiki/Cardinality.