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# cardinality

# Cardinality

*Cardinality* is a notion of the size of a set which does not rely on numbers. It is a relative notion. For instance, two sets may each have an infinite number of elements, but one may have a greater cardinality. That is, in a sense, one may have a “more infinite” number of elements. See Cantor diagonalization for an example of how the reals have a greater cardinality than the natural numbers.

The formal definition of cardinality rests upon the notion mappings between sets:

###### Cardinality.

and

###### Cardinality.

It can be shown that if $|A|\geq|B|$ and $|B|\geq|A|$ then $|A|=|B|$. This is the SchrÃ¶der-Bernstein Theorem.

Equality of cardinality is variously called *equipotence*, *equipollence*, *equinumerosity*, or *equicardinality*.
For $|A|=|B|$, we would say that “$A$ is *equipotent* to $B$”,
“$A$ is *equipollent* to $B$”, or “$A$ is *equinumerous* to $B$”.

An equivalent definition of cardinality is

###### Cardinality (alt. def.).

The cardinality of a set $A$ is the unique cardinal number $\kappa$ such that $A$ is equinumerous with $\kappa$. The cardinality of $A$ is written $|A|$.

# Results

Some results on cardinality:

1. $A$ is equipotent to $A$.

2. If $A$ is equipotent to $B$, then $B$ is equipotent to $A$.

3. If $A$ is equipotent to $B$ and $B$ is equipotent to $C$, then $A$ is equipotent to $C$.

###### Proof.

Respectively:

1. The identity function on $A$ is a bijection from $A$ to $A$.

2. If $f$ is a bijection from $A$ to $B$, then $f^{{-1}}$ exists and is a bijection from $B$ to $A$.

3. If $f$ is a bijection from $A$ to $B$ and $g$ is a bijection from $B$ to $C$, then $f\circ g$ is a bijection from $A$ to $C$.

∎

# Example

The set of even integers $2\mathbb{Z}$ has the same cardinality as the set of integers $\mathbb{Z}$: if we define $f\colon 2\mathbb{Z}\to\mathbb{Z}$ such that $f(x)=\frac{x}{2}$, then $f$ is a bijection, and therefore $|2\mathbb{Z}|=|\mathbb{Z}|$.

## Mathematics Subject Classification

03E10*no label found*

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## Comments

## Do proper classes have cardinalities?

This entry defines cardinality, as a concept, only for sets. Can it be extended to deal with proper classes? If so, do they vary in cardinality?

## Re: Do proper classes have cardinalities?

The cardinality |X| of a set X is most usefully taken to be the cardinal number with which X is equinumerous. The arithmetic of cardinal numbers then provides a means of calculating the cardinalities of sets constructed via set operations from the cardinalities of their constituents. Note, for example, that |{0, 1}| = 2, and |A|^|B| = |{f | f:B -> A}|. The set {f | f:X -> {0, 1}} is just the set of characteristic functions of subsets of X. Thus, the power set of X has cardinality 2^|X|.

There are no proper classes in ZFC, so questions about them simply don't arise. In the class theories NBG and MK all proper classes are equinumerous with the class of ordinal numbers so, by abusing conventional usage, all proper classes could be said to have the same cardinality. But why bother?