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In mathematics, two sets are almost disjoint 〔Kunen, K. (1980), "Set Theory; an introduction to independence proofs", North Holland, p. 47〕〔Jech, R. (2006) "Set Theory (the third millennium edition, revised and expanded)", Springer, p. 118〕 if their 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. 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 (1 ) and (2 ) are almost disjoint, because their intersection is the finite set . However, the unit interval (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''''i'' be a set. Then the collection of sets 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 is an almost disjoint collection consisting of more than one set, 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Almost disjoint sets」の詳細全文を読む スポンサード リンク
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