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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.
In mathematics, the disjoint union (or discriminated union) of the sets A and B is the set formed from the elements of A and B labelled (indexed) with the name of the set from which they come. So, an element belonging to both A and B appears twice in the disjoint union, with two different labels.
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 ...
Pairwise generally means "occurring in pairs" or "two at a time." Pairwise may also refer to: Pairwise disjoint; Pairwise independence of random variables; Pairwise comparison, the process of comparing two entities to determine which is preferred; All-pairs testing, also known as pairwise testing, a software testing method.
To define the join, form a relation on the blocks A of α and the blocks B of ρ by A ~ B if A and B are not disjoint. Then α ∨ ρ {\displaystyle \alpha \vee \rho } is the partition in which each block C is the union of a family of blocks connected by this relation.
The subdomains together must cover the whole domain; often it is also required that they are pairwise disjoint, i.e. form a partition of the domain. [5] In order for the overall function to be called "piecewise", the subdomains are usually required to be intervals (some may be degenerated intervals, i.e. single points or unbounded intervals).
For example, the closed intervals [0, 1] and [1, 2] are almost disjoint, because their intersection is the finite set {1}. 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.
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} .