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This definition would allow pairwise disjoint families of sets to have repeated copies of the same set. According to an alternative definition, each two sets in the family must be disjoint; repeated copies are not allowed. The same two definitions can be applied to an indexed family of sets: according to the first definition, every two distinct ...
In category theory the disjoint union is defined as a coproduct in the category of sets. As such, the disjoint union is defined up to an isomorphism, and the above definition is just one realization of the coproduct, among others. When the sets are pairwise disjoint, the usual union is another realization of the coproduct.
The set of those translates partitions the circle into a countable collection of pairwise disjoint sets, which are all pairwise congruent. Since X is not measurable for any rotation-invariant countably additive finite measure on S , finding an algorithm to form a set from selecting a point in each orbit requires that one add the axiom of choice ...
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.
In combinatorics, a laminar set family is a set family in which each pair of sets are either disjoint or related by containment. [1] [2] Formally, a set family {S 1, S 2, ...} is called laminar if for every i, j, the intersection of S i and S j is either empty, or equals S i, or equals S j. Let E be a ground-set of elements.
A partition of a set is defined as a family of nonempty, pairwise disjoint subsets of whose union is . For example, = because the 3-element set {,,} can be ...
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 sets E, O, U are pairwise-disjoint. Proof: U is disjoint from E and O by definition. To prove that E and O are disjoint, suppose that some vertex v has both an even-length alternating path to an unmatched vertex u 1, and an odd-length alternating path to an unmatched vertex u 2.