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The abstraction of cardinality as a number is evident by 3000 BCE, in Sumerian mathematics and the manipulation of numbers without reference to a specific group of things or events. [ 6 ] From the 6th century BCE, the writings of Greek philosophers show hints of the cardinality of infinite sets.
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 cardinality of any infinite ordinal number is an aleph number. Every aleph is the cardinality of some ordinal. The least of these is its initial ordinal. Any set whose cardinality is an aleph is equinumerous with an ordinal and is thus well-orderable. Each finite set is well-orderable, but does not have an aleph as its cardinality.
As a consequence, the cardinality of the real numbers, which is the same as that of the power set of the integers, is strictly larger than the cardinality of the integers; see Cardinality of the continuum for details. The theorem is named for Georg Cantor, who first stated and proved it at the end of the 19th century.
This inverse has a special structure, making the principle an extremely valuable technique in combinatorics and related areas of mathematics. As Gian-Carlo Rota put it: [ 6 ] "One of the most useful principles of enumeration in discrete probability and combinatorial theory is the celebrated principle of inclusion–exclusion.
In mathematics, a multiset (or bag, or ... The cardinality or "size" of a multiset is the sum of ... The multiplicative formula allows the definition of multiset ...
The set of real algebraic numbers is countably infinite (assign to each formula its Gödel number.) So the cardinality of the real algebraic numbers is . Furthermore, the real algebraic numbers and the real transcendental numbers are disjoint sets whose union is .
In the mathematical field of set theory, the continuum means the real numbers, or the corresponding (infinite) cardinal number, denoted by . [ 1 ] [ 2 ] Georg Cantor proved that the cardinality c {\displaystyle {\mathfrak {c}}} is larger than the smallest infinity, namely, ℵ 0 {\displaystyle \aleph _{0}} .