Disjoint sets

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.

[3] In topology, there are various notions of separated sets with more strict conditions than disjointness.

Similarly, in a metric space, positively separated sets are sets separated by a nonzero distance.

However, a collection of sets may have an empty intersection without being disjoint.

[2] For instance, the three sets { {1, 2}, {2, 3}, {1, 3} } have an empty intersection but are not disjoint.

Also the empty family of sets is pairwise disjoint.

[6] A Helly family is a system of sets within which the only subfamilies with empty intersections are the ones that are pairwise disjoint.

For instance, the closed intervals of the real numbers form a Helly family: if a family of closed intervals has an empty intersection and is minimal (i.e. no subfamily of the family has an empty intersection), it must be pairwise disjoint.

[8] Every partition can equivalently be described by an equivalence relation, a binary relation that describes whether two elements belong to the same set in the partition.

[8] Disjoint-set data structures[9] and partition refinement[10] are two techniques in computer science for efficiently maintaining partitions of a set subject to, respectively, union operations that merge two sets or refinement operations that split one set into two.

[12] For instance two sets may be made disjoint by replacing each element by an ordered pair of the element and a binary value indicating whether it belongs to the first or second set.

[13] For families of more than two sets, one may similarly replace each element by an ordered pair of the element and the index of the set that contains it.