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Preliminaries
Published in J. Tinsley Oden, Leszek F. Demkowicz, Applied Functional Analysis, 2017
J. Tinsley Oden, Leszek F. Demkowicz
The union of two sets A and B is the set of all elements x that belong to A or B. The union of A and B is denoted by A∪B $ A\cup B $ and, using the notation introduced previously, AA∪B=defx:x∈Aorx∈B $$ AA\cup B \ \mathop {=}\limits ^\mathrm{def}\ \left\{ x:\ x\in A \text{ or} x\in B \right\} $$
General introduction
Published in Adedeji B. Badiru, Handbook of Industrial and Systems Engineering, 2013
Associative law for intersection A∩(B∩C)=(A∩B)∩CDistributive law for unionA∪(B∩C)=(A∪B)∩(A∪C)Distributive law for intersectionA∩(B∩C)=(A∩B)∪(A∩C)
Functions
Published in Janet Woodcock, Software Engineering Mathematics, 1988
We have defined union and intersection as binary operators: the two arguments they take and the result that they produce are of the same type. We can be more adventurous and define generalised operators that take an arbitrary number of arguments and produce a result, all of the same type. We can collect the arguments into a set and thereby avoid being too specific about how many arguments there are. The fact that set union is both commutative and associative means that the order in which the arguments are united is unimportant. The distributed union of a set of sets SS is the set of all elements which are members of some member of SS. Examples are
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.