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...l [[cardinality]] is a '''transcendence basis''' for ''E''/''F'', and this cardinality is the '''transcendence degree''' or '''transcendence dimension''' of ''E''

{{r|cardinality}} -->

{{r|Cardinality}}

{{r|Cardinality}}

...ring for which any two bases for a free module over the ring have the same cardinality is said to have the [[invariant dimension property]].

The '''order''' of a group is just its [[cardinality]] as a set. The connexion between the two is that the order of an element

...s. Some of the intuitive notions associated with size do not carry over to cardinality, and some do in certain set theories but not in others. The cardinality of a set ''X'' then, will be an object associated with ''X'', which we deno

A special case of size is [[cardinality]],

i.e., its [[cardinality]] (the number of its elements) is a natural [[number]].

== Cardinality ==

In the context of counting, '''zero''' is the smallest [[cardinality]], the number of elements in the empty set.

...ine the '''rank''' ρ(''A'') of a subset ''A'' of ''E'' to be the maximum [[cardinality]] of an independent subset of ''A''. The rank satisfies the following

The ''rank'' of a free group is the [[cardinality]] of a generating set. A subgroup of a free group is again free, but the r

...l number|transcendental]]. The non-existence of a subset of the reals with cardinality strictly between that of the integers and the reals is known as the [[conti ...real numbers themselves. The set of [[hyperreal number]]s <!--is equal in cardinality to '''R''' and also--> satisfies the same first order sentences as '''R'''.

...[[one-to-one correspondence]] with its [[power set]]: equivalently, the [[cardinality]] of a set is strictly less than that of its power set. The argument proce

==== Cardinality ==== ...4,6,8, ...}. One is a subset of the other. Nevertheless they have the same cardinality, as is shown by the correspondence mapping ''n'' in the former set to 2''n'

==== Cardinality ==== ...4,6,8, ...}. One is a subset of the other. Nevertheless they have the same cardinality, as is shown by the correspondence mapping ''n'' in the former set to 2''n'

...ilar" is interpreted as [[Bijective function#Bijections and the concept of cardinality|equinumerous]].

...ilar" is interpreted as [[Bijective function#Bijections and the concept of cardinality|equinumerous]].

...a finite set contains is called that set's [[cardinality]]. The concept of cardinality can also be applied to infinite sets, though the concept is less intuitive, ...lacing bars around the name of the set. For example, one would express the cardinality of the above set as such: