Disjoint sets

In mathematics, two sets are said to be disjoint sets if they have no element in common. Equivalently, two disjoint sets are sets whose intersection is the empty set. For example, {1, 2, 3} and {4, 5, 6} are disjoint sets, while {1, 2, 3} and {3, 4, 5} are not disjoint. A collection of two or more sets is called disjoint if any two distinct sets of the collection are disjoint. This definition of disjoint sets can be extended to families of sets and to indexed families of sets. By definition, a collection of sets is called a family of sets (such as the power set, for example). In some sources this is a set of sets, while other sources allow it to be a multiset of sets, with some sets repeated. An is by definition is a set-valued function (that is, it is a function that assigns a set to every element in its domain) whose domain is called its (and elements of its domain are called ). There are two subtly different definitions for when a family of sets is called pairwise disjoint. According to one such definition, the family is disjoint if each two sets in the family are either identical or disjoint. This definition would allow pairwise disjoint families of sets to have repeated copies of the same set. According to an alternative definition, each two sets in the family must be disjoint; repeated copies are not allowed. The same two definitions can be applied to an indexed family of sets: according to the first definition, every two distinct indices in the family must name sets that are disjoint or identical, while according to the second, every two distinct indices must name disjoint sets. For example, the family of sets { {0, 1, 2}, {3, 4, 5}, {6, 7, 8}, ... } is disjoint according to both definitions, as is the family { {..., −2, 0, 2, 4, ...}, {..., −3, −1, 1, 3, 5} } of the two parity classes of integers. However, the family with 10 members has five repetitions each of two disjoint sets, so it is pairwise disjoint under the first definition but not under the second. Two sets are said to be almost disjoint sets if their intersection is small in some sense.