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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~\}.}
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.
In the case where K is a simplex, the two simplex faces formed by the maximum and minimum points of g must then be two disjoint faces whose images have a nonempty intersection. This same general statement, when applied to a hypersphere instead of a simplex, gives the Borsuk–Ulam theorem , that ƒ must map two opposite points of the sphere to ...
In geometry, the hyperplane separation theorem is a theorem about disjoint convex sets in n-dimensional Euclidean space.There are several rather similar versions. In one version of the theorem, if both these sets are closed and at least one of them is compact, then there is a hyperplane in between them and even two parallel hyperplanes in between them separated by a gap.
The algebra of sets is the set-theoretic analogue of the algebra of numbers. Just as arithmetic addition and multiplication are associative and commutative, so are set union and intersection; just as the arithmetic relation "less than or equal" is reflexive, antisymmetric and transitive, so is the set relation of "subset".
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 .
The algebra of sets is an interpretation or model of Boolean algebra, with union, intersection, set complement, U, and {} interpreting Boolean sum, product, complement, 1, and 0, respectively. The properties below are stated without proof , but can be derived from a small number of properties taken as axioms .