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There are two ways to define the "cardinality of a set": The cardinality of a set A is defined as its equivalence class under equinumerosity. A representative set is designated for each equivalence class. The most common choice is the initial ordinal in that class. This is usually taken as the definition of cardinal number in axiomatic set theory.
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 task is then to find a minimum cardinality subset of left-vertices that has a non-trivial intersection with each of the right-vertices, which is precisely the hitting set problem. In the field of computational geometry, a hitting set for a collection of geometrical objects is also called a stabbing set or piercing set. [5]
A set is countable if it can be enumerated, that is, if there exists an enumeration of it. Otherwise, it is uncountable. For example, the set of the real numbers is uncountable. A set is finite if it can be enumerated by means of a proper initial segment {1, ..., n} of the natural numbers, in which case, its cardinality is n.
Assuming the existence of an infinite set N consisting of all natural numbers and assuming the existence of the power set of any given set allows the definition of a sequence N, P(N), P(P(N)), P(P(P(N))), … of infinite sets where each set is the power set of the set preceding it. By Cantor's theorem, the cardinality of each set in this ...
Cardinal functions are widely used in topology as a tool for describing various topological properties. [4] [5] Below are some examples.(Note: some authors, arguing that "there are no finite cardinal numbers in general topology", [6] prefer to define the cardinal functions listed below so that they never take on finite cardinal numbers as values; this requires modifying some of the definitions ...
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 ...
The HyperLogLog has three main operations: add to add a new element to the set, count to obtain the cardinality of the set and merge to obtain the union of two sets. Some derived operations can be computed using the inclusion–exclusion principle like the cardinality of the intersection or the cardinality of the difference between two HyperLogLogs combining the merge and count operations.