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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".
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| ==Definition==
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| 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
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| :<math>\left|A\cap B\right| < \infty.</math>
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| (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.
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| 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".
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| 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'',
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| :<math>A_i \ne A_j \quad \Rightarrow \quad \left|A_i \cap A_j\right| < \infty.</math>
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| 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:
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| :<math>\bigcap_{i\in I} A_i < \infty.</math>
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| However, the converse is not true—the intersection of the collection
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| :<math>\{\{1, 2, 3,\ldots\}, \{2, 3, 4,\ldots\}, \{3, 4, 5,\ldots\},\ldots\}</math>
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| is empty, but the collection is ''not'' almost disjoint; in fact, the intersection of ''any'' two distinct sets in this collection is infinite.
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| ==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):
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| *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
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| :<math>\left|A\cap B\right| < \kappa.</math> | |
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| :The case of κ = 1 is simply the definition of [[disjoint sets]]; the case of
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| :<math>\kappa = \aleph_0</math>
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| :is simply the definition of almost disjoint given above, where the intersection of ''A'' and ''B'' is finite.
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| *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
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| :<math>m(A\cap B) = 0.</math>
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| *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''.
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| ==References==
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| {{Reflist}}
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| {{DEFAULTSORT:Almost Disjoint Sets}}
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| [[Category:Set families]]
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Adrianne Le is the business name my parents gave me to but you can cellphone me anything you just like. 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
My homepage - clash of clans hacker