## Cardinality

The cardinality of a set written is a measure of the &quot;number of elements of the set&quot;. The cardinality of a finite set is equal to the number of elements of the set. For infinite sets cardinality is measured using either bijections/injections or cardinal numbers.

Two sets have the same cardinality if there exists an injective (one to one) and surjective (onto) function from to The set of odd natural numbers has the same cardinality as the set of natural numbers with the function from to given by All infinite countable sets have the same cardinality as the smallest set with infinite cardinality
If there exists an injective function from to then If there is an injective but no bijective function then the cardinality of is greater than the cardinality of The cardinality of the set of real numbers is greater than the cardinality of for this reason.
The cardinality of a set can be defined in terms of equivalence relations. The equivalence classof a set A under this relation then consists of all those sets which have the same cardinality as A.

WE check that the equivalence axioms are satisfied:

1. since the identity function from to itself is bijective and surjective.

2. there exists an injective and surjective function and is well defined because is injective and surjective hence 3. there exists an injective and surjective function and there exists an injective and surjective function so is an injective and surjective function from to so Any set with cardinality less than is said to be a finite set.

Any set that has the same cardinality as is a countably infinite set.

Any set with cardinality greater than is said to be uncountable.

The cardinality of the continuum ( ) is greater than that of the natural numbers ( ); that is, there are more real numbers than whole numbers  Because for any open interval in there exists a bijection (eg from that interval onto the cardinality of is equal to the cardinality of any open interval in Cardinal arithmetic can be used to show not only that the number of points in is equal to the number of points in any open interval of and to the number of points on a plane and, indeed, in for any   That the cardinal number of is equal to the cardinal number of may be demonstrated with the use of space filling curves, which demonstrate a bijection from to The set of all subsets of and the set of all functions from to both have cardinality  