mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, two sets are almost disjoint Kunen, K. (1980), "Set Theory; an introduction to independence proofs", North Holland, p. 47Jech, R. (2006) "Set Theory (the third millennium edition, revised and expanded)", Springer, p. 118 if their

intersection In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their ...

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 :

\left, A \cap B\ < \infty.

(Here, ', ''X'', ' denotes the

cardinality The thumb is the first digit of the hand, next to the index finger. When a person is standing in the medical anatomical position (where the palm is facing to the front), the thumb is the outermost digit. The Medical Latin English noun for thum ...

of ''X'', and '< ∞' means 'finite'.) For example, the closed intervals , 1and , 2are almost disjoint, because their intersection is the finite set . However, the unit interval , 1and the set of

rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (for example, The set of all ...

s 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 ''A''_''i'' be a set. Then the collection of sets is almost disjoint if for any ''i'' and ''j'' in ''I'', :

A_i \ne A_j \quad \implies \quad \left, A_i \cap A_j\ < \infty.

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 is an almost disjoint collection consisting of more than one set, then clearly its intersection is finite: :

\bigcap_ A_i < \infty.

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. The possible cardinalities of a maximal almost disjoint family (commonly referred to as a MAD family) on the set

\omega

of the

natural number In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...

s has been the object of intense study. The minimum infinite such

cardinal Cardinal or The Cardinal most commonly refers to * Cardinalidae, a family of North and South American birds **''Cardinalis'', genus of three species in the family Cardinalidae ***Northern cardinal, ''Cardinalis cardinalis'', the common cardinal of ...

is one of the classical cardinal characteristics of the continuum.

Other meanings

Sometimes "almost disjoint" is used in some other sense, or in the sense of

measure theory In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude (mathematics), magnitude, mass, and probability of events. These seemingl ...

or topological category. Here are some alternative definitions of "almost disjoint" that are sometimes used (similar definitions apply to infinite collections): *Let κ be any cardinal number. Then two sets ''A'' and ''B'' are almost disjoint if the cardinality of their intersection is less than κ, i.e. if ::

\left, A\cap B\ < \kappa.

:The case of κ = 1 is simply the definition of disjoint sets; the case of ::

\kappa = \aleph_0

:is simply the definition of almost disjoint given above, where the intersection of ''A'' and ''B'' is finite. *Let ''m'' be a

complete measure In mathematics, a complete measure (or, more precisely, a complete measure space) is a measure space in which every subset of every null set is measurable (having measure zero). More formally, a measure space (''X'', Σ, ''μ'') is comp ...

on a

measure space A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the -algebra) and the method that ...

''X''. Then two subsets ''A'' and ''B'' of ''X'' are almost disjoint if their intersection is a null-set, i.e. if ::

m(A\cap B) = 0.

*Let ''X'' be a

topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...

. Then two subsets ''A'' and ''B'' of ''X'' are almost disjoint if their intersection is meagre in ''X''.

References

{{DEFAULTSORT:Almost Disjoint Sets Families of sets