Introduction
The concept of a Fuzzy Logic is one that it is very easy for the ill-informed to dismiss as trivial and/or insignificant. It refers not to a fuzziness of logic but instead to a logic of fuzziness, or more specifically to the logic of fuzzy sets. Those that examined Lotfi A. Zadeh's concept more closely found it to be useful for dealing with real world phenomena. From a strictly mathematical point of view the concept of a Fuzzy Set is a brilliant generalization of the classical notion of a Set. Now the concept of a Fuzzy Set is well established as an important and practical construct for modeling. Moreover, Zadeh's formulation makes one realize how artificial is the classical black-white formulation of Aristotelian logic (Is A or Is Not-A). In a world of shades of gray a black-white dichotomy involves an unnecessary arbitrariness, an artificiality imposed upon that world.
The purpose of the material here is to present the mathematical structure of the concept of Fuzzy Sets. This generalization is achieved by way of the concept of the characteristic function for a set.
Classical Set Theory Formulated
in Terms of Characteristic Functions
One way of defining a set A is in terms of its characteristic function μA(x). A point x belongs to set A if and only if μA(x)=1. A characteristic function is a function from some universal set U to the binary set {0,1}.
The set operations of union, intersection and complementation are defined in terms of characteristic functions as follows.
- Union:
μA∪B(x) = max(μA(x),μB(x))
- Intersection:
μA∩B(x) = min(μA(x),μB(x))
- Complement:
μnot A(x) = 1-μA(x))
The other set theory constructs that are essential are:
- Set Inclusion:
A ⊂ B if and only if ∀x (for all x)
μA(x)=1 implies μB(x)=1
- Set Equality:
A = B if and only if ∀x (for all x)
μA(x)=μB(x).
A Fuzzy Set as a Generalization of a Regular (Crisp) Set
As indicated above a characteristic function is a mapping from the universal set U to the set {0,1}. A fuzzy set is defined in terms of a membership function which is a mapping from the universal set U to the interval [0,1]. A characteristic function is a special case of a membership function and a regular set (a.k.a a crisp set) is a special case of a fuzzy set. Thus the concept of a fuzzy set is a natural generalization of the concept of standard set theory.
It remains to be proven whether the standard operations of standard set theory; i.e., union, intersection and complementation, have proper analogues in fuzzy set theory.
Fuzzy Set Theory in Terms of Membership Functions
A membership function is a function from a universal set U to the interval [0,1]. A fuzzy set A is defined by its membership function φA over U.
The operation of union, intersection and complementation are defined exactly the same as they are for standard sets in terms of the characteristic function; i.e.;
- Union:
φA∪B(x) = max(φA(x),φB(x))
- Intersection:
φA∩B(x) = min(φA(x),φB(x))
- Complement:
φnot A(x) = 1-φA(x))
Set inclusion and set equality have a natural definition for fuzzy sets; i.e.,
- Set Inclusion:
A ⊂ B if and only if ∀x (for all x)
φA(x)≤φB(x)
- Set Equality:
A = B if and only if ∀x (for all x)
φA(x)=φB(x).
Of course any definitions can be posited; the question is whether the corresponding theorems that hold in standard set theory hold in fuzzy set theory. With the above definitions most standard set theory theorems carry over into fuzzy set theory.
Basic Properties of Sets and Set Operations
Some of the more important elementary theorems of standard set theory are:
- Associativity of Union:
A ∪ (B ∪ C) = (A ∪ B) ∪ C
- Commutativity of Union:
A ∪ B = B ∪ A
- Associativity of Intersection:
A ∩ (B ∩ C) = (A ∩ B) ∩ C
- Commutativity of Intersection:
A ∩ B = B ∩ A
- Distributivity of Union with respect to Intersection:
A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
- Distributivity of Intersection with respect to Union:
A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
- Existence of a null set Φ such that for any set A
A ∪ Φ = A and A ∩ Φ = Φ
- Reflexity of Complementation:
(Ac)c = A
- De Morgan's Laws:
(A ∪ B)c = Ac ∩ Bc
(A ∩ B)c = Ac ∪ Bc
In the analysis below let φA, φB and φC be the membership functions for the fuzzy sets A, B, and C respectively. Furthermore, for any element of the universal set p,
x = φA(p),
y = φB(p)
and z = φC(p).
The associativity and commutativity of fuzzy set union and intersection follow from the definition and the associativity and commutativity of the maximum and minimum functions; i.e.,
max(x,max(y,z)) = max(max(x,y),z)
min(x,min(y,z)) = min(min(x,y),z)
The distributivity properties also follow from properties of the maximum and minimum functions but the proof is a bit longer.
The right-hand side of the first distributivity relation is (A ∩ B) ∪ (A ∩ C) which for fuzzy sets involves the evaluation of w = max(min(x,y),min(x,z)). If x is less either y or z then w = x. If x is between y
The right-hand side of the second distributivity relation is (A ∪ B) ∩ (A ∪ C) which requires the evaluation of w = min(max(x,y),max(x,z)). As in the case of the previous distributivity relation the various cases can be evaluated. If x is greater than either y or z then w=x. If x is between y and z for y
The reflexity of complementation is easily established.
1 - (1 - x) = x
The null set for fuzz sets is the fuzzy set Φ for which the membership function is zero for all elements.
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