Almost Disjoint Sets - Definition

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 and are almost disjoint, because their intersection is the finite set {1}. However, the unit interval 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 {A_i : i in I} 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 {A_i} is an almost disjoint collection, 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.