关于有限论域上模糊度定理的一个注记

A Note on Theorem for Fuzzy Degree over Finite Universe

设F(U)为有限论域U={u1,u2,…,un}上的模糊幂集.令D:F(U)→[0,1],AaD(A)=g(∑n i=1ci fi(A(ui))),其中ci>0,fi:[0,1]→[0,+∞)满足(1)fi(x)=fi(1-x),(2)fi(0)=0,(3)fi在[0,0.5]上严格递增.又设a=∑n i=1ci fi(0,5),且g:[0,a]→[0,1]严格递增,g(0)=0,g(a)=1.某教材中的定理断定具有上述性质的映射D为F(U)上的模糊度函数.本文构造出具有上述性质的D,但D不满足模糊度函数定义中的可加性条件,故不为F(U)上的模糊度函数.本文进一步指出,若将g的定义域扩展为[0,2a],并再假定g是可加的,则可使D为F(U)上的模糊度函数.

Let F（U）be a fuzzy power set of a finite universe of discourse U ={u1,u2,…,un },and D：F（U）→[0,1],AaD（A）=g（∑i=1^ n ci fi（A（ui）））,where ci 0,and fi ：[0,1]→[0,＋∞）satisfies（1）fi（x）= fi（1-x）,（2）fi（0）= 0,（3）fiis strictly increasing on[0,0.5].Suppose that a=∑ i=1^n ci fi（0,5）and g：[0,a]→[0,1]is strictly increasing,and g（0）=0,g（a）=1.A theorem in a textbook asserts that a mapping Dpossessing above properties is a fuzzy degree function on F（U）.In this paper,a mapping D possessing above properties but fail to satisfy the additive condition in the definition of fuzzy degree function is constructed.It is pointed out further that gcan be a fuzzy degree function if it is additive and its domain is extended to[0,2a].