Search results
Results from the WOW.Com Content Network
A bijective function, f: X → Y, from set X to set Y demonstrates that the sets have the same cardinality, in this case equal to the cardinal number 4. Aleph-null, the smallest infinite cardinal. In mathematics, a cardinal number, or cardinal for short, is what is commonly called the number of elements of a set.
Cardinal arithmetic can be used to show not only that the number of points in a real number line is equal to the number of points in any segment of that line, but that this is equal to the number of points on a plane and, indeed, in
In set theory, a regular cardinal is a cardinal number that is equal to its own cofinality. More explicitly, this means that κ {\displaystyle \kappa } is a regular cardinal if and only if every unbounded subset C ⊆ κ {\displaystyle C\subseteq \kappa } has cardinality κ {\displaystyle \kappa } .
The use of Scott's trick for cardinal numbers shows how the method is typically employed. The initial definition of a cardinal number is an equivalence class of sets, where two sets are equivalent if there is a bijection between them.
The von Neumann cardinal assignment is a cardinal assignment that uses ordinal numbers. For a well-orderable set U , we define its cardinal number to be the smallest ordinal number equinumerous to U , using the von Neumann definition of an ordinal number.
The cardinality of a set X is essentially a measure of the number of elements of the set. [1] Equinumerosity has the characteristic properties of an equivalence relation (reflexivity, symmetry, and transitivity): [1] Reflexivity Given a set A, the identity function on A is a bijection from A to itself, showing that every set A is equinumerous ...
Notably, is the first uncountable cardinal number that can be demonstrated within Zermelo–Fraenkel set theory not to be equal to the cardinality of the set of all real numbers: For any natural number , we can consistently assume that =, and moreover it is possible to assume that is as least as large as any cardinal number we like.
The definition of a finite set is given independently of natural numbers: [3] Definition: A set is finite if and only if any non empty family of its subsets has a minimal element for the inclusion order. Definition: a cardinal n is a natural number if and only if there exists a finite set of which the cardinal is n. 0 = Card (∅) 1 = Card({A ...