Search results
Results from the WOW.Com Content Network
If | X | ≤ | Y | and | Y | ≤ | X |, then | X | = | Y |. This holds even for infinite cardinals, and is known as Cantor–Bernstein–Schroeder theorem . Sets with cardinality of the continuum include the set of all real numbers, the set of all irrational numbers and the interval [ 0 , 1 ] {\displaystyle [0,1]} .
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 .
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.
Generally speaking, X Y is the set of all functions from Y to X and | X Y | = | X | | Y |. Cantor's diagonal argument shows that the power set of a set (whether infinite or not) always has strictly higher cardinality than the set itself (or informally, the power set must be larger than the original set).
The oldest definition of the cardinality of a set X (implicit in Cantor and explicit in Frege and Principia Mathematica) is as the set of all sets that are equinumerous with X: this does not work in ZFC or other related systems of axiomatic set theory because this collection is too large to be a set, but it does work in type theory and in New ...
In fact, the cardinality of ℘ (), by definition , is equal to . This can be shown by providing one-to-one mappings in both directions between subsets of a countably infinite set and real numbers, and applying the Cantor–Bernstein–Schroeder theorem according to which two sets with one-to-one mappings in both directions have the same ...
The search engine that helps you find exactly what you're looking for. Find the most relevant information, video, images, and answers from all across the Web.
This larger set consists of the elements (x 1, x 2, x 3, ...), where each x n is either m or w. [3] Each of these elements corresponds to a subset of N—namely, the element (x 1, x 2, x 3, ...) corresponds to {n ∈ N: x n = w}. So Cantor's argument implies that the set of all subsets of N has greater cardinality than N.