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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.
There are two notions often used when referring to cardinality: one which compares sets directly using bijections and injections, and another which uses cardinal numbers. [2] The cardinality of a set may also be called its size, when no confusion with other notions of size is possible. [a]
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.
compact cardinal A cardinal number that is uncountable and has the property that any collection of sets of that cardinality has a subcollection of the same cardinality with a non-empty intersection. complement (of a set) The set containing all elements not in the given set, within a larger set considered as the universe. complete 1.
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 } .
is a finite set with five elements. The number of elements of a finite set is a natural number (possibly zero) and is called the cardinality (or the cardinal number) of the set. A set that is not a finite set is called an infinite set. For example, the set of all positive integers is infinite:
Cardinal functions are widely used in topology as a tool for describing various topological properties. [2] [3] Below are some examples.(Note: some authors, arguing that "there are no finite cardinal numbers in general topology", [4] prefer to define the cardinal functions listed below so that they never taken on finite cardinal numbers as values; this requires modifying some of the ...
Each ordinal has an associated cardinal, its cardinality, obtained by simply forgetting the order. Any well-ordered set having that ordinal as its order type has the same cardinality. The smallest ordinal having a given cardinal as its cardinality is called the initial ordinal of that cardinal.