Almost disjoint sets
Encyclopedia
In mathematics
, two sets are almost disjoint if their intersection
is small in some sense; different definitions of "small" will result in different definitions of "almost disjoint".
(Here, '|X|' denotes the cardinality of X, and '< ∞' means 'finite'.) For example, the closed intervals [0, 1] and [1, 2] are almost disjoint, because their intersection is the finite set {1}. However, the unit interval [0, 1] and the set of rational numbers Q are not almost disjoint, because their intersection is infinite.
This definition 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 almost disjoint. Often the prefix "pairwise" is dropped, and a pairwise almost disjoint collection is simply called "almost disjoint".
Formally, let I be an index set
, and for each i in I, let Ai be a set. Then the collection of sets {Ai : i in I} is almost disjoint if for any i and j in I,
For example, the collection of all lines through the origin in R2
is almost disjoint, because any two of them only meet at the origin. If {Ai} is an almost disjoint collection, then clearly its intersection is finite:
However, the converse is not true—the intersection of the collection
is empty, but the collection is not almost disjoint; in fact, the intersection of any two distinct sets in this collection is infinite.
or topological category
. Here are some alternative definitions of "almost disjoint" that are sometimes used (similar definitions apply to infinite collections):
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, two sets are almost disjoint if their intersection
Intersection (set theory)
In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B , but no other elements....
is small in some sense; different definitions of "small" will result in different definitions of "almost disjoint".
Definition
The most common choice is to take "small" to mean finite. In this case, two sets are almost disjoint if their intersection is finite, i.e. if(Here, '|X|' denotes the cardinality of X, and '< ∞' means 'finite'.) For example, the closed intervals [0, 1] and [1, 2] are almost disjoint, because their intersection is the finite set {1}. However, the unit interval [0, 1] and the set of rational numbers Q are not almost disjoint, because their intersection is infinite.
This definition 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 almost disjoint. Often the prefix "pairwise" is dropped, and a pairwise almost disjoint collection is simply called "almost disjoint".
Formally, let I be an index set
Index set
In mathematics, the elements of a set A may be indexed or labeled by means of a set J that is on that account called an index set...
, and for each i in I, let Ai be a set. Then the collection of sets {Ai : i in I} is almost disjoint if for any i and j in I,
For example, the collection of all lines through the origin in R2
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
is almost disjoint, because any two of them only meet at the origin. If {Ai} is an almost disjoint collection, then clearly its intersection is finite:
However, the converse is not true—the intersection of the collection
is empty, but the collection is not almost disjoint; in fact, the intersection of any two distinct sets in this collection is infinite.
Other meanings
Sometimes "almost disjoint" is used in some other sense, or in the sense of measure theoryMeasure (mathematics)
In mathematical analysis, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume...
or topological category
Baire space
In mathematics, a Baire space is a topological space which, intuitively speaking, is very large and has "enough" points for certain limit processes. It is named in honor of René-Louis Baire who introduced the concept.- Motivation :...
. Here are some alternative definitions of "almost disjoint" that are sometimes used (similar definitions apply to infinite collections):
- Let κ be any cardinal numberCardinal numberIn mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality of sets. The cardinality of a finite set is a natural number – the number of elements in the set. The transfinite cardinal numbers describe the sizes of infinite...
. Then two sets A and B are almost disjoint if the cardinality of their intersection is less than κ, i.e. if
- The case of κ = 1 is simply the definition of disjoint sets; the case of
- is simply the definition of almost disjoint given above, where the intersection of A and B is finite.
- Let m be a complete measure on a measure space X. Then two subsets A and B of X are almost disjoint if their intersection is a null-set, i.e. if
- Let X be a topological spaceTopological spaceTopological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
. Then two subsets A and B of X are almost disjoint if their intersection is meagreBaire spaceIn mathematics, a Baire space is a topological space which, intuitively speaking, is very large and has "enough" points for certain limit processes. It is named in honor of René-Louis Baire who introduced the concept.- Motivation :...
in X.