First Course on Fuzzy Theory and Applications by Lee K.H.

By Lee K.H.

This simple textbook provides an simply obtainable advent to Fuzzy idea and its purposes. It presents easy and urban ideas of the sector in a self-contained, condensed and comprehensible type. This "First path on Fuzzy idea and functions" contains a number of examples, descriptive illustrations and figures of the fundamental options, in addition to workouts on the finish of every bankruptcy. This publication can be utilized in introductory graduate and undergraduate classes in Fuzziness and gentle Computing and is recommendable to scholars, scientists, engineers, or pros within the box for self-study.

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Example text

If we express this mapping as f, y is called the image of x which is denoted as f(x) R {(x, y) | x  A, y  B, y f(x)} or f : A o B So we might say ran(R) is the set gathering of these f(x) ran(R) f(A) {f(x) | x  A} 56 3. Fuzzy Relation and Composition A B R dom(R) ran(R) Fig. 3. Domain and range A x1 B f y1 x2 y2 x3 y3 Fig. 4. 3 Characteristics of Relation We can summarize the properties of relation as follows : (1) One-to-many (Fig. 5) R is said to be one-to-many if  x  A, y1, y2  B (x, y1)  R, (x, y2)  R (2) Surjection (may-to-one) R is said to be a surjection if f(A) B or ran(R)  y  B,  x  A, y f(x) B.

3 1 a) Compare the support of each set. b) What is the normalized fuzzy set? c) Find the level set of each set. 3. 4 Determine whether the following fuzzy sets are convex or not. 6 Compute the scalar cardinality and the fuzzy cardinality for each of the following fuzzy set. 7 Show the following set is convex. ­0  2 1 ¯(1  ( x  10) ) P A ( x) ® x d 10 x ! 9. Chapter 2. THE OPERATION OF FUZZY SET In the previous section, we have studied complement, union and intersection operations of fuzzy sets.

And in other cases symbolȉis used for t-norm, while symbol A for t-conorm, xȉy : t-norm, x A y : t-conorm 46 2. The Operation of Fuzzy Set t-norm is sometimes called s-norm. In practical applications, sometimes we could not determine which operator is appropriate to the application. In that case, we put the symbol * as a tentative operator. When we know that a t-norm operator can be used without selecting a specific operator, we put the symbolȉ. In this manner we can refine a system step by step.

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