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Venn diagram showing the union of sets A and B as everything not in white. In combinatorics, the inclusion–exclusion principle is a counting technique which generalizes the familiar method of obtaining the number of elements in the union of two finite sets; symbolically expressed as
A Venn diagram is a widely used diagram style that shows the logical relation between sets, popularized by John Venn (1834–1923) in the 1880s. The diagrams are used to teach elementary set theory, and to illustrate simple set relationships in probability, logic, statistics, linguistics and computer science.
The simplest way to introduce cardinals is to add a primitive notion, Card(), and an axiom of cardinality to ZF set theory (without axiom of choice). [2] Axiom of cardinality: The sets A and B are equinumerous if and only if Card(A) = Card(B)
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.
In set theory, Kőnig's theorem states that if the axiom of choice holds, I is a set, and are cardinal numbers for every i in I, and < for every i in I, then <. The sum here is the cardinality of the disjoint union of the sets m i, and the product is the cardinality of the Cartesian product.
The relation of having the same cardinality is called equinumerosity, and this is an equivalence relation on the class of all sets. The equivalence class of a set A under this relation, then, consists of all those sets which have the same cardinality as A. There are two ways to define the "cardinality 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.
The goal of a cardinal assignment is to assign to every set A a specific, unique set that is only dependent on the cardinality of A. This is in accordance with Cantor 's original vision of cardinals: to take a set and abstract its elements into canonical "units" and collect these units into another set, such that the only thing special about ...