Search results
Results from the WOW.Com Content Network
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 ...
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 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".
The simple theorems in the algebra of sets are some of the elementary properties of the algebra of union (infix operator: ∪), intersection (infix operator: ∩), and set complement (postfix ') of sets. These properties assume the existence of at least two sets: a given universal set, denoted U, and the empty set, denoted {}.
Two curves that overlap represent sets that intersect, that have common elements; the zone inside both curves represents the set of elements common to both sets (the intersection of the sets). A curve completely within the interior of another is a subset of it. Venn diagrams are a more restrictive form of Euler diagrams.
Intersection [e] If R and S are relations over X then R ∩ S = { (x, y) | xRy and xSy} is the intersection relation of R and S. The identity element of this operation is the universal relation. For example, "is a lower card of the same suit as" is the intersection of "is a lower card than" and "belongs to the same suit as". Composition [e]
The relaxed intersection of m sets corresponds to the classical intersection between sets except that it is allowed to relax few sets in order to avoid an empty intersection. This notion can be used to solve constraints satisfaction problems that are inconsistent by relaxing a small number of constraints .
Let be a set and a nonempty family of subsets of ; that is, is a subset of the power set of . Then is said to have the finite intersection property if every nonempty finite subfamily has nonempty intersection; it is said to have the strong finite intersection property if that intersection is always infinite.