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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.
The quantity—a cardinal number—is associated with the set, and is invariant under the process of counting. An identity is an equation that remains true for all values of its variables. There are also inequalities that remain true when the values of their variables change.
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 ...
The continuum hypothesis says that =, i.e. is the smallest cardinal number bigger than , i.e. there is no set whose cardinality is strictly between that of the integers and that of the real numbers. The continuum hypothesis is independent of ZFC , a standard axiomatization of set theory; that is, it is impossible to prove the continuum ...
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.
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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.