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Again, the integral notation should not be algebraically interpreted
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Membership Functions
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The membership function is the essence of fuzzy sets A membership function, also referred to as the characteristic function of the fuzzy set, de nes the fuzzy set The function is used to associate a degree of membership of each of the elements of the domain to the corresponding fuzzy set Two-valued sets are also characterized by a membership function For example, consider the domain X of all oating-point numbers in the range [0, 100] De ne the crisp set A X of all oating-point numbers in the range [10, 50] Then, the membership function for the crisp set A is represented in Figure 201 All x [10, 50] have A (x) = 1, while all other oating-point numbers have A (x) = 0 Membership functions for fuzzy sets can be of any shape or type as determined by experts in the domain over which the sets are de ned While designers of fuzzy sets
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202 Membership Functions
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Figure 201 Illustration of Membership Function for Two-Valued Sets
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tall(x)
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Figure 202 Illustration of tall Membership Function have much freedom in selecting appropriate membership functions, these functions must satisfy the following constraints: A membership function must be bounded from below by 0 and from above by 1 The range of a membership function must therefore be [0, 1] For each x X, A (x) must be unique That is, the same element cannot map to di erent degrees of membership for the same fuzzy set Returning to the tall fuzzy set, a possible membership function can be de ned as (also illustrated in Figure 202) if length(x) < 15m 0 (length(x) 15m) 20m if 15m length(x) 20m tall(x) = (205) 1 if length(x) > 20m Now, assume that a person has a length of 175m, then A (175) = 05 While the tall membership function above used a discrete step function, more complex discrete and continuous functions can be used, for example:
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456 Triangular functions (refer to Figure 203(a)), de ned as 0 x min if x min if x ( min , ] min A (x) = max x max if x ( , max ) 0 if x max Trapezoidal functions (refer to Figure 203(b)), de ned as 0 x min if x min if x [ min , 1 ) 1 min A (x) = max x max 2 if x ( 2 , max ) 0 if x max -membership functions, de ned as A x = 0 2 1 e (x ) if x if x >
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20 Fuzzy Sets
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S-membership functions, de ned as 0 2 x min 2 max min A (x) = 1 2 x max max min 1
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if x min if x ( min , ]
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if x ( , max ) if x max
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Logistic function (refer to Figure 203(c)), de ned as A (x) = Exponential-like function, de ned as A (x) = with > 1 Gaussian function (refer to Figure 203(d)), de ned as A (x) = e (x )
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It is the task of the human expert of the domain to de ne the function that captures the characteristics of the fuzzy set
203 Fuzzy Operators
Fuzzy Operators
As for crisp sets, relations and operators are de ned for fuzzy sets Each of these relations and operators are de ned below For this purpose let X be the domain, or universe, and A and B are fuzzy sets de ned over the domain X Equality of fuzzy sets: For two-valued sets, sets are equal if the two sets have exactly the same elements For fuzzy sets, however, equality cannot be concluded if the two sets have the same elements The degree of membership of elements to the sets must also be equal That is, the membership functions of the two sets must be the same Therefore, two fuzzy sets A and B are equal if and only if the sets have the same domain, and A (x) = B (x) for all x X That is, A = B Containment of fuzzy sets: For two-valued sets, A B if all the elements of A are also elements of B For fuzzy sets, this de nition is not complete, and the degrees of membership of elements to the sets have to be considered Fuzzy set A is a subset of fuzzy set B if and only if A (x) B (x) for all x X That is, A B Figure 204 shows two membership functions for which A B Complement of a fuzzy set (NOT): The complement of a two-valued set is simply the set containing the entire domain without the elements of that set For fuzzy sets, the complement of the set A consists of all the elements of set A, but the membership degrees di er Let A denote the complement of set A Then, for all x X, A (x) = 1 A (x) It also follows that A A = and A A = X Intersection of fuzzy sets (AND): The intersection of two-valued sets is the set of elements occurring in both sets Operators that implement intersection are referred to as t-norms The result of a t-norm is a set that contain all the elements of the two fuzzy sets, but with degree of membership that depends on the speci c t-norm A number of t-norms have been used, of which the minoperator and the product operator are the most popular If A and B are two fuzzy sets, then Min-operator: A B (x) = min{ A (x), B (x)}, x X Product operator: A B (x) = A (x) B (x), x X The di erence between the two operations should be noted Taking the product of membership degrees is a much stronger operator than taking the minimum, resulting in lower membership degrees for the intersection It should also be noted that the ultimate result of a series of intersections approaches 00, even if the degrees of memberships to the original sets are high Other t-norms are [676], A B (x) =
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