generally, a collection of any sets whatsoever is called a family of sets, set family, or a set system. Additionally, a family of sets may be defined as a...

sources this is a set of sets, while other sources allow it to be a multiset of sets, with some sets repeated. An indexed family of sets ( A i ) i ∈ I ,...

Sets are uniquely characterized by their elements; this means that two sets that have precisely the same elements are equal (they are the same set)....

Branch of mathematics concerning probability Ring of sets – Family closed under unions and relative complements Set function – Function from sets to numbers...

two different notions of a ring of sets, both referring to certain families of sets. In order theory, a nonempty family of sets R {\displaystyle {\mathcal...

In mathematics, specifically set theory, the Cartesian product of two sets A and B, denoted A × B, is the set of all ordered pairs (a, b) where a is in...

algebra of sets, not to be confused with the mathematical structure of an algebra of sets, defines the properties and laws of sets, the set-theoretic...

if and only if every family of closed subsets with F.I.P. is fixed. Filter (set theory) – Family of sets representing "large" sets Mac Lane, Saunders;...

consequences; in particular, by defining different collections of sets containing 0 (distinct from the sets (−ε, ε)), one may find different results regarding the...

combinatorics, a laminar set family is a set family in which each pair of sets are either disjoint or related by containment. Formally, a set family {S1, S2, ...}...

uncountable family of subsets of the natural numbers indexed by the real numbers. Let I {\displaystyle I} and X {\displaystyle X} be sets and f {\displaystyle...

sets have been released per year, in addition to various spin-off products. Magic has made three types of sets since Alpha and Beta: base/core sets,...

collection is typically called an indexed family, often written as {Aj}j∈J. An enumeration of a set S gives an index set J ⊂ N {\displaystyle J\subset \mathbb...

{P} (S)} , or 2S. Any subset of P(S) is called a family of sets over S. If S is the set {x, y, z}, then all the subsets of S are {} (also denoted ∅ {\displaystyle...

The sets in P are said to exhaust or cover X. See also collectively exhaustive events and cover (topology). The intersection of any two distinct sets in...

(X)} will denote the power set of X . {\displaystyle X.} A subset of a power set is called a family of sets (or simply, a family) where it is over X {\displaystyle...

mathematics, the cylinder sets form a basis of the product topology on a product of sets; they are also a generating family of the cylinder σ-algebra. Given...

to finite families of sets that can be infinite. However, the situation of having an infinite number of sets while allowing infinite sets is not allowed...

non-measurable sets do exist; an example is the Vitali sets. The first part of the definition states that the subset E {\displaystyle E} of the real numbers...

ordered sets are directed sets (contrast partially ordered sets, which need not be directed). Join-semilattices (which are partially ordered sets) are directed...

countable family of closed sets of uniqueness is a set of uniqueness. There exists an example of two sets of uniqueness whose union is not a set of uniqueness...

holds for any two sets, then it also holds for any finite number of sets, namely, the function value on the union of k disjoint sets (where k is a finite...

parameters Family of sets, a collection of sets Indexed family, a family where each element can be given an index Normal family, a collection of continuous...

indexed family of sets is a collection of sets, indexed by a fixed set; equivalently, a function from a fixed index set to the class of sets. A diagram...

compactness in terms of closed sets; this is its most prominent application. Other applications include proving that certain perfect sets are uncountable,...

paradox concerns the impossibility of a set of sets, whose members are all sets that do not contain themselves. If such a set could exist, it could neither...

Mathematical construction of a set with an equivalence relation Transversal (combinatorics) – Set that intersects every one of a family of sets Devlin 2004, p. 122...

allow one to use whichever of these two notions is more technically convenient. There is a certain preorder on families of sets, denoted by ≤ , {\displaystyle...

extends to any collection of sets. A collection of sets is pairwise almost disjoint or mutually almost disjoint if any two distinct sets in the collection are...

combinatorics, a Helly family of order k is a family of sets in which every minimal subfamily with an empty intersection has k or fewer sets in it. Equivalently...