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Devising and Synthesis of NEMS and MEMS
Published in Sergey Edward Lyshevski, Nano- and Micro-Electromechanical Systems, 2018
We consider sets A and B in ℝn and ℝm. If a = {a1, a2,…,an−1, an} and b = {b1, b2, bm−1, bm} are elements of A and B, then one writes a ∈ A and b ∈ B. If a and b are not elements of A and B, we write a ∉ A and b ∉ B. The union of two sets A and B is denoted as A ⋃ B, and the intersection is denoted by A ⋂ B. The common set operations are the union, intersection, and complement. The union of A and B is the set of all elements that are either in A or B (or both). Thus, A ∪ B = {x | x ∈ A or x ∈ B}. The union in the Venn diagram is shown in Figure 5.6. The intersection of A and B is the set of all elements that are common to A and B, i.e., A ⋃ B = {x | x ∈ A and x ∈ B}. If A is a subset of S, then A’ is the complement of A in S, the set of all elements of S not in A (see Figure 5.6). Figure 5.6 also illustrates the Venn diagram if B ⊆ A.
Information Granularity, Information Granules, and Granular Computing
Published in Witold Pedrycz, Granular Computing, 2018
where A(x) stands for a value of the characteristic function of set A at point x. With the emergence of digital technologies, interval mathematics has appeared as an important discipline encompassing many applications. A family of sets defined in a universe of discourse X is denoted by P(X). Well-known set operations—union, intersection, and complement are the three fundamental constructs supporting a manipulation on sets. In terms of the characteristic functions, they result in the following expressions, () (A∩B)(x)=min(A(x),B(x))(A∩B)(x)=max(A(x),B(x))A¯(x)=1−A(x)
Discrete Mathematics
Published in Dan Zwillinger, CRC Standard Mathematical Tables and Formulas, 2018
Basic set operations include:The union of the sets A and B, denoted A ∪ B, are all the elements that are inA or are in B. The intersection of the sets A and B, denoted A ∩ B, are all the elements in both A and B.
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
Step 1. Determining the partition set for both intersection and union (or any other fuzzy set operations) is the same. Firstly, consider the intervals , then the set is a partition of . To prove this, we have three situations: If , then .If , then .If , , then .
A feature selection method with feature ranking using genetic programming
Published in Connection Science, 2022
Guopeng Liu, Jianbin Ma, Tongle Hu, Xiaoying Gao
GP has flexible representation ability. In general, the GP-based feature selection method takes original features as the terminal set and terminal nodes as the selected features. However, some researchers have researched other representations. Hunt et al. (2012) developed GP-based hyper-heuristics for adding and removing features. Each individual generated by GP is a series of operations on the feature set. Ribeiro et al. (2012) used four feature selection criteria (information gain, chi-square, correlation and odds ration) as the terminal set of GP and three set operations (intersection, union and difference) as the function set. Viegas et al. (2018) proposed a GP-based feature selection method for skewed dataset and validated the method on four high-dimensional datasets. Papa et al. (2017) used binary strings as individual's terminal nodes, where 1 means selecting a certain feature and 0 means without selecting. AND, OR and XOR are regarded as non-terminal nodes. The output of the individual is also a binary string, which contains the selected features.
The usefulness and application of fuzzy logic and fuzzy AHP in the materials finishing industry
Published in Transactions of the IMF, 2020
Two important operations in fuzzy logic, particularly in control applications, are union and intersection. The union of two sets is the set which contains all elements common to both sets as well as those contained in only one or other of them.This is the equivalent of Boolean OR, whereas the equivalent of Boolean AND in fuzzy logic operations is set intersection.Thus, using union and intersection set operations allows fuzzy logic to implement Boolean OR & AND logic rule-based inference. The union and intersection of two fuzzy sets A and B carried out using straightforward max–min compositions (Mathematica software) is shown graphically in Figure 3. For special purposes, there are several other possible ways that can be used to accomplish these operations including the use of Hamacher, Frank or Yager formulas. These formulas indicate different ways in which the membership grades for corresponding elements in the two sets are combined but for the great majority of fuzzy set union and intersection operations, the standard max–min compositions are used.