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.
The cardinality of a set $A$ is greater than or equal to the cardinality of a set $B$ if there is a onetoone function (an injection^{}) from $B$ to $A$. Symbolically, we write $A\ge B$.
and
Cardinality.
It can be shown that if $A\ge B$ and $B\ge A$ then $A=B$. This is the SchrÃÂ¶derBernstein 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 $A$.
This definition of cardinality makes use of a special class of numbers, called the cardinal numbers. This highlights the fact that, while cardinality can be understood and defined without appealing to numbers, it is often convenient and useful to treat cardinality in a “numeric” manner.
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 $f:2\mathbb{Z}\to \mathbb{Z}$ such that $f(x)=\frac{x}{2}$, then $f$ is a bijection, and therefore $2\mathbb{Z}=\mathbb{Z}$.
Title  cardinality 
Canonical name  Cardinality 
Date of creation  20130322 12:00:42 
Last modified on  20130322 12:00:42 
Owner  yark (2760) 
Last modified by  yark (2760) 
Numerical id  25 
Author  yark (2760) 
Entry type  Definition 
Classification  msc 03E10 
Synonym  size 
Related topic  OrderGroup 
Related topic  GeneralizedContinuumHypothesis 
Related topic  CardinalNumber 
Related topic  DedekindInfinite 
Defines  equipotence 
Defines  equipotent 
Defines  equicardinality 
Defines  equipollence 
Defines  equipollent 
Defines  equinumerosity 