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:
The cardinality of a set is greater than or equal to the cardinality of a set if there is a one-to-one function (an injection) from to . Symbolically, we write .
It can be shown that if and then . This is the SchrÃÂ¶der-Bernstein Theorem.
Equality of cardinality is variously called equipotence, equipollence, equinumerosity, or equicardinality. For , we would say that “ is equipotent to ”, “ is equipollent to ”, or “ is equinumerous to ”.
An equivalent definition of cardinality is
Cardinality (alt. def.).
The cardinality of a set is the unique cardinal number such that is equinumerous with . The cardinality of is written .
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.
Some results on cardinality:
is equipotent to .
If is equipotent to , then is equipotent to .
If is equipotent to and is equipotent to , then is equipotent to .
The identity function on is a bijection from to .
If is a bijection from to , then exists and is a bijection from to .
If is a bijection from to and is a bijection from to , then is a bijection from to .
The set of even integers has the same cardinality as the set of integers : if we define such that , then is a bijection, and therefore .
|Date of creation||2013-03-22 12:00:42|
|Last modified on||2013-03-22 12:00:42|
|Last modified by||yark (2760)|