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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. U:[0,1] u [0,1] o [0,1] this union function calculates the membership degree of union A୶i from those of A and B. PAB(x) = U[PA(x), PB(x)] this union function should obey next axioms. (Axiom U1) (Axiom U2) U(0,0) = 0, U(0,1) = 1, U(1,0) = 1, U(1,1) = 1 so this union function follows properties of union operation of crisp sets (boundary condition).

1 0 x1 x2 x3 x4 Fig. 11. 1 0 x1 x2 x3 x4 Fig. 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’.

20 1. 1, there are 5 elements in the D-cut set. 1| = 5. 6, there are 2 at D=1. 6), (2,1)}. 6 Subset of fuzzy set Suppose there are two fuzzy sets A and B. When their degrees of membership are same, we say “A and B are equivalent”. That is, A = B iff PA(x) = PB(x), x  X If PA(x) z PB(x) for any element, then A z B. 15). PA(x) d PB(x), x  X This relation is expressed as A  B. We call that A is a subset of B. In addition, if the next relation holds, A is a proper subset of B. PA(x)  PB(x), x  X This relation can be written as AB iff A  B and A z B.