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Preliminaries
Published in Hugo D. Junghenn, Principles of Analysis, 2018
A family {Ai:i∈I} $ \{A_i: i\in \mathfrak I \} $ of sets is disjoint if Ai∩Aj=∅ $ A_i\cap A_j= \emptyset $ whenever i≠j $ i \not = j $ . In this case, the union ⋃i∈IAi $ \bigcup _{i\in \mathfrak I } A_i $ is said to be disjoint. A partition of a set X is a collection of nonempty, disjoint sets whose union is X.
Relations
Published in Rowan Garnier, John Taylor, Discrete Mathematics, 2020
As we mentioned in chapter 3, there is a close connection between partitions of a set and equivalence relations on the set. Recall that a partition of a set A is a family of subsets which are pairwise disjoint and whose union is all of A (see definition 3.4). Suppose R is an equivalence relation on A. We can form subsets by grouping together in the same subset all elements which are related. We shall see that the properties of the equivalence relation guarantee that the subsets formed in this way form a partition of A. These subsets are called the ‘ equivalence classes ' of the relation which we now define formally.
Strait fuzzy sets, strait fuzzy rough sets and their similarity measures-based decision making systems
Published in International Journal of Systems Science, 2023
Akın Osman Atagün, Hüseyin Kamacı
As can be understood from Definition 3.2, many strait fuzzy sets can be produced for one fuzzy set. Already, one of the facilitating features of strait fuzzy sets is that the intervals can be adjusted for the situations required by the problem. As is known, a partition of a set U corresponds to an equivalence relation on U. Classical fuzzy sets require a direct evaluation with a value. In strait fuzzy sets, it is possible to evaluate using values that are equivalent to the value obtained through classical fuzzy sets.
Centralized collaborative production scheduling with evaluation of a practical order-merging strategy
Published in International Journal of Production Research, 2023
Liang Tang, Huanying Han, Zhen Tan, Ke Jing
The solution space of the order processing problem with K single orders (in which, there are np single orders of type p, p = 1, … , P, as shown as Figure 3) is the same as the combination of the partitions of the subsets with n1, … , nP elements. In mathematics, a partition of a set Ω is a division of Ω into pairwise disjoint nonempty subsets whose union is Ω. These subsets are both collectively exhaustive and mutually exclusive with respect to the set being partitioned (Brualdi 2004). Here, Ω is the set of all K single orders, and Ωp is the subset of np single orders. Let X denote an arbitrary solution of the order processing problem of K single orders. Then the total solutions are combinations of solutions of the subsets Ωp. The number of feasible solutions S(K) can be represented as follows: where s(np, k) is the number of solutions of the order processing problem of subset Ωp, which is a Stirling number of the second type (Weisstein 2010). Since the feasible solutions are composed of merged processing methods and the non-merged processing method, the number of merged processing methods F is S(K)–1. We show the solutions of a 5-order problem (where orders 1, 2, and 3 belong to type 1, and orders 4 and 5 belong to type 2) as an example. All 10 solutions are listed as follows: 1,2,3,4,5, 1,2,3,4,5, 1,2,3,4,5, 1,2,3,4,5, 1,3,2,4,5, 1,3,2,4,5, 2,3,1,4,5, 2,3,1,4,5, 1,2,3,4,5, 1,2,3,4,5. Obviously, unlike the subsets Ωp, this is not simply a partition problem of the set Ω = 1, 2, 3, 4, 5, but the combination of a partition of subsets Ω1 = 1, 2, 3 and Ω2 = 4, 5.