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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 ...
Set packing is a classical NP-complete problem in computational complexity theory and combinatorics, and was one of Karp's 21 NP-complete problems. Suppose one has a finite set S and a list of subsets of S. Then, the set packing problem asks if some k subsets in the list are pairwise disjoint (in other words, no two of them share an element).
The set system consists of pairwise disjoint sets , …, with sizes ,,, …, respectively, as well as two additional disjoint sets ,, each of which contains half of the elements from each . On this input, the greedy algorithm takes the sets S k , … , S 1 {\displaystyle S_{k},\ldots ,S_{1}} , in that order, while the optimal solution consists ...
This article lists mathematical properties and laws of sets, involving the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion. It also provides systematic procedures for evaluating expressions, and performing calculations, involving these operations and relations.
A partition of a set is defined as a family of nonempty, pairwise disjoint subsets of whose union is . For example, B 3 = 5 {\displaystyle B_{3}=5} because the 3-element set { a , b , c } {\displaystyle \{a,b,c\}} can be partitioned in 5 distinct ways:
Then consider ,, …, to be a maximal collection of pairwise disjoint sets (that is, is the empty set unless =, and every set in intersects with some ). Because we assumed that W {\displaystyle W} had no sunflower of size r {\displaystyle r} , and a collection of pairwise disjoint sets is a sunflower, t < r {\displaystyle t<r} .
Disjoint-set data structures model the partitioning of a set, for example to keep track of the connected components of an undirected graph. This model can then be used to determine whether two vertices belong to the same component, or whether adding an edge between them would result in a cycle.
A partition of a set S is a set of non-empty, pairwise disjoint subsets of S, called "parts" or "blocks", whose union is all of S.Consider a finite set that is linearly ordered, or (equivalently, for purposes of this definition) arranged in a cyclic order like the vertices of a regular n-gon.