Basic Concepts of Fuzzy Sets

Given a fuzzy set A defined on X, the largest value of the membership function A for all x X is called the height of A and it is usually denoted by h A). When h(.A)=l, A is called a normal fuzzy set otherwise, it is called a subnormal fuzzy set. 

Among the most important concepts associated with fuzzy sets are the concepts of an a-cut and a strong a-cut. Given a fuzzy set A defined on X and a number a in the unit interval [0,1], the a-cut of A, denoted by °A, is the crisp set that consists of all elements of A whose membership degrees in A are greater than or equal to a that is, 

The strong a-cut ° A has a similar meaning but the condition greater than or equal to is replaced with the condition greater than. Hence, 

The introduced concepts are illustrated by the trapezoidal membership function A in Fig. 2. This function is defined on the closed interval [a,b] of real numbers (i.e., X = [a,b]), which may represent the range of values of a physical variable. 

Given any arbitrary fuzzy set A and any two numbers a, a2 e [0,1] such that a, a2, it is obvious, by the given definitions, that the set inclusions 

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A detailed review of the basic concepts of fuzzy sets can be found in other chapters of this volume. Here only the specific notations and the fuzzy set concepts most relevant to the molecular shape problem are reviewed, followed by a simple proof for a special fuzzy set generalization of the Hausdorff distance, motivated by the quantum chemical properties of fuzzy electronic densities of molecules. 

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