Search results
Results From The WOW.Com Content Network
In set theory, the union (denoted by ∪) of a collection of sets is the set of all elements in the collection. [1] It is one of the fundamental operations through which sets can be combined and related to each other. A nullary union refers to a union of zero ( ) sets and it is by definition equal to the empty set.
So the intersection of the empty family should be the universal set (the identity element for the operation of intersection), [4] but in standard set theory, the universal set does not exist. However, when restricted to the context of subsets of a given fixed set X {\displaystyle X} , the notion of the intersection of an empty collection of ...
The algebra of sets is the set-theoretic analogue of the algebra of numbers. Just as arithmetic addition and multiplication are associative and commutative, so are set union and intersection; just as the arithmetic relation "less than or equal" is reflexive, antisymmetric and transitive, so is the set relation of "subset".
This article lists mathematical properties and laws of sets, involving the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion. It also provides systematic procedures for evaluating expressions, and performing calculations, involving these operations and relations.
The double-counted elements are those in the intersection of the two sets and the count is corrected by subtracting the size of the intersection. The inclusion-exclusion principle, being a generalization of the two-set case, is perhaps more clearly seen in the case of three sets, which for the sets A , B and C is given by
If some power set ℘ is partially ordered in the usual way (by ) then joins are unions and meets are intersections; in symbols, = = (where the similarity of these symbols may be used as a mnemonic for remembering that denotes the join/supremum and denotes the meet/infimum [note 1]).
The union of sets is distributive over intersection, and intersection is distributive over union. ... is a metalogical symbol representing "can be replaced in a ...
The Cantor–Bendixson theorem states that any Polish space can be written as the union of a countable set and a perfect set. Because any G δ subset of a Polish space is again a Polish space, the theorem also shows that any G δ subset of a Polish space is the union of a countable set and a set that is perfect with respect to the induced topology.