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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.

5. 0 32 2. 3 Fuzzy Partition Let A be a crisp set in universal set X and A be a complement set of A. The conditions A z ‡ and A˸  result in couple ( A , A ) which decomposes X into 2 subsets. Definition (Fuzzy partition) In the same manner, consider a fuzzy set satisfying A z ‡ and A˸ U. The pair ( A , A ) is defined as fuzzy partition. Usually, if m subsets are defined in X, m-tuple (A1,A2,…,An) holding the following conditions is called a fuzzy partition. 1 Axioms for Union Function In general sense, union of A and B is specified by a function of the form.

12. 4 By using standard complement and intersection operations, the difference operation would be simple. 7), (x3, 0), (x4, 0)} Ƒ B A Fig. 13. 2 x1 x2 x3 x4 Fig. 14. Simple difference A – B 42 2. 1 x1 x2 x3 x4 Fig. 15. 15). 3 Distance in Fuzzy Set The concept ‘distance’ is designated to describe the difference. 16). Measures for distance are defined in the following. 5 Other Operations in Fuzzy Set 43 Hamming distance contains usual mathematical senses of ‘distance’. , |X| = n, the relative Hamming distance is, G(A, B) = 1 d(A, B) n might rename Hamming distance as ‘symmetrical distance’ and written as below by using operator ’, x  X , P A’B ( x) | P A ( x)  P B ( x) | This operator doesn’t hold distributivity.

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A nondegeneracy result for a nonlinear elliptic equation by Grossi M.


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