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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 ...
Repeat step three until there is a new row with one more number than the previous row (do step 3 until = +) The number on the left hand side of a given row is the Bell number for that row. (,) Here are the first five rows of the triangle constructed by these rules:
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.
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 constructive mathematics, "not empty" and "inhabited" are not equivalent: every inhabited set is not empty but the converse is not always guaranteed; that is, in constructive mathematics, a set that is not empty (where by definition, "is empty" means that the statement () is true) might not have an inhabitant (which is an such that ).
The optimization version of the problem, maximum set packing, asks for the maximum number of pairwise disjoint sets in the list. It is a maximization problem that can be formulated naturally as an integer linear program , belonging to the class of packing problems .
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.
By definition of wandering sets and since preserves , would thus contain a countably infinite union of pairwise disjoint sets that have the same -measure as . Since it was assumed μ ( X ) < ∞ {\displaystyle \mu (X)<\infty } , it follows that A {\displaystyle A} is a null set, and so all wandering sets must be null sets.