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Appendix
Fuzzy Operator Formulas
The following is a list of formulas used to operate on the membership grades of fuzzy set A for selected general operations on fuzzy sets. In the following formulas, xn represents the new membership grades after applying the operators, and yn represents the original membership grades for the elements of fuzzy set A
Concentrate
 Dilate
 Intensify Contrast
 Normalize
 Complement
Standard
Sugeno[ ], w  (-1,  )
Yager[w], w (0, )
 Intersection Formulas
The following is a list of the formulas used for various intersections between two fuzzy sets, A and B. The following formulas indicate how the membership grades for corresponding elements in fuzzy sets A and B should be combined. In the formulas, a represents the membership grade for an element in fuzzy set A, and b represents the membership grade for an element in fuzzy set B.
Standard
Hamacher[v], v (0, )
Frank[s], s (0, )  s  1.
Yager[w], w (0, )
DuboisPrade[], [0, 1]
Dombi[], (0, )
Weber[ ], (-1, )
Yu[], (-1 , )
 Union Formulas
The following is a list of the formulas used for various unions between two fuzzy sets, A and B. The following formulas indicate how the membership grades for corresponding elements in fuzzy sets A and B should be combined. In the formulas, a represents the membership grade for an element in fuzzy set A, and b represents the membership grade for the corresponding element in fuzzy set B.
Standard
Hamacher[v], v (0, )
Frank[s], s (0, ) s 1
Yager[w], w (0, )
DuboisPrade[], [0, 1]
Dombi[], (0, )
Weber[], (-1 , )
Yu[], (-1 , )
 Averaging Formulas
Following is a list of the formulas for taking the various averages used in this package. The averaging operations are denoted by the letter h, and  represent the membership grades for corresponding elements in the n fuzzy sets being averaged.
Arithmetic Mean
 Geometric Mean
 Harmonic Mean
 Generalized Mean[], (- , )
 Miscellaneous Formulas
Gaussian Fuzzy Sets
 where m is the mean, s is the width, and x is the element.
Bell Fuzzy Sets
 where c is the center, crossover points are at c ± w, a slope at the crossover points of s/2w, and x is the element.
Sigmoid Fuzzy Sets
 where s controls the slope at crossover point c, and x is the element.
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