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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".
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]
An example of how intersecting sets define a graph. In graph theory, an intersection graph is a graph that represents the pattern of intersections of a family of sets.Any graph can be represented as an intersection graph, but some important special classes of graphs can be defined by the types of sets that are used to form an intersection representation of them.
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 {}.
More generally, in set theory, the intersection of sets is defined to be the set of elements which belong to all of them. Unlike the Euclidean definition, this does not presume that the objects under consideration lie in a common space. It simply means the overlapping area of two or more objects or geometries.