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	In [[mathematics]], two [[Set (mathematics)\|sets]] are '''almost disjoint''' <ref name = "kunen">Kunen, K. (1980), "Set Theory; an introduction to independence proofs", North Holland, p. 47</ref><ref name = "jech">Jech, R. (2006) "Set Theory (the third millennium edition, revised and expanded)", Springer, p. 118</ref>if their [[intersection (set theory)\|intersection]] 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 set\|finite]]. In this case, two sets are almost disjoint if their intersection is finite, i.e. if~~

	~~:<math>\left\|A\cap B\right\| < \infty.</math>~~		Adrianne Le is the business name my parents gave me to but you can [http://Browse.Deviantart.com/?q=cellphone cellphone] me anything you just like. [http://mondediplo.com/spip.php?page=recherche&recherche=Vermont Vermont] has always been simple home and I find it irresistible every day living in this case. As a girl what I actually like is to toy croquet but I are unable make it my profession really. Filing offers been my profession a few time and I'm doing pretty good financially. You can find my net site here: http://prometeu.net<br><br>My homepage - [http://prometeu.net clash of clans hacker]

	~~(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 ''A''<sub>''i''</sub> be a set. Then the collection of sets {''A''<sub>''i''</sub>~~ : ~~''i'' in ''I''} is almost disjoint if for any ''i'' and ''j'' in ''I'',~~

	~~:<math>A_i \ne A_j \quad \Rightarrow \quad \left\|A_i \cap A_j\right\| < \infty.<~~/~~math>~~

	~~For example, the collection of all lines through the origin in [[Euclidean space\|'''R'''<sup>2<~~/~~sup>]] is almost disjoint, because any two of them only meet at the origin~~. ~~If {''A''<sub>''i''</sub>} is an almost disjoint collection consisting of more than one set, then clearly its intersection is finite:~~

	~~:<math>\bigcap_{i\in I} A_i < \infty~~.~~</math>~~

	~~However, the converse is not true—the intersection of the collection~~
	~~:<math>\{\{1, 2, 3,\ldots\}, \{2, 3, 4,\ldots\}, \{3, 4, 5,\ldots\},\ldots\}<~~/~~math>~~
	~~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 (mathematics)\|measure theory]] or [[Baire space\|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~~

	:~~<math>\left\|A\cap B\right\| < \kappa~~.</~~math>~~

	~~:The case of~~ &~~kappa;~~ = ~~1 is simply the definition of [[disjoint sets~~]~~]; the~~ case of

	~~:<math>\kappa = \aleph_0</math>~~

	:is ~~simply the definition of almost disjoint given above, where the intersection of ''A'' and ''B'' is finite~~.

	*Let ''m'' be a [[measure theory\|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

	:<~~math~~>~~m(A\cap B) = 0.~~<~~/math~~>

	*Let ''X'' be a [~~[topological space]]~~. ~~Then two subsets ''A'' and ''B''~~ of ~~''X'' are almost disjoint if their intersection is [[Baire space\|meagre]] in ''X''.~~

	~~==References==~~
	~~{{Reflist}}~~

	~~{{DEFAULTSORT:Almost Disjoint Sets}}~~
	~~[[Category:Set families]~~]

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