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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".
A set of polygons in an Euler diagram This set equals the one depicted above since both have the very same elements.. In mathematics, a set is a collection of different [1] things; [2] [3] [4] these things are called elements or members of the set and are typically mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other ...
A set is described by listing elements separated by commas, or by a characterizing property of its elements, within braces { }. [5] Since sets are objects, the membership relation can relate sets as well, i.e., sets themselves can be members of other sets. A derived binary relation between two sets is the subset relation, also called set inclusion.
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 {}.
Additionally, while a collection of less than two sets is trivially disjoint, as there are no pairs to compare, the intersection of a collection of one set is equal to that set, which may be non-empty. [2] For instance, the three sets { {1, 2}, {2, 3}, {1, 3} } have an empty intersection but are not disjoint. In fact, there are no two disjoint ...
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.