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Two disjoint sets. In set theory in mathematics and formal logic, 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. [1] 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 ...
One set is said to intersect another set if . Sets that do not intersect are said to be disjoint . The power set of X {\displaystyle X} is the set of all subsets of X {\displaystyle X} and will be denoted by ℘ ( X ) = def { L : L ⊆ X } . {\displaystyle \wp (X)~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\{~L~:~L\subseteq X~\}.}
The Kneser graph ,, with a vertex for each 2-element subset of the 5-element set {1,2,3,4,5} and an edge for each pair of disjoint subsets. According to the Erdős–Ko–Rado theorem, the independent sets in this graph have at most four vertices.
A pair of pants as a plane domain (in blue, with the boundary in red) A pair of pants is any surface that is homeomorphic to a sphere with three holes, which formally is the result of removing from the sphere three open disks with pairwise disjoint closures. Thus a pair of pants is a compact surface of genus zero with three boundary components.
A disjoint union of an indexed family of sets (:) is a set , often denoted by , with an injection of each into , such that the images of these injections form a partition of (that is, each element of belongs to exactly one of these images). A disjoint union of a family of pairwise disjoint sets is their union.
That is, for any sets ,, and , one has = () = () Inside a universe , one may define the complement of to be the set of all elements of not in . Furthermore, the intersection of A {\displaystyle A} and B {\displaystyle B} may be written as the complement of the union of their complements, derived easily from De Morgan's laws : A ∩ B = ( A c ...
However, the unit interval [0, 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 ...
In computability theory, two disjoint sets of natural numbers are called computably inseparable or recursively inseparable if they cannot be "separated" with a computable set. [1] These sets arise in the study of computability theory itself, particularly in relation to Π 1 0 {\displaystyle \Pi _{1}^{0}} classes .