Properties of fuzzy sets

ROPERTIES OF A FUZZY SET

 
Recall that a fuzzy (sub)set A of a set of all possible (feasible, relevant) elements with respect to a fuzzy concept, say, X = {x}.

Then: A of X Þ A Ì X
                              ~
Þ {(x, m_A (x)}
Similarly: B of x Þ B Ì X Þ {(x, m_B (x)}
                                  ~

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P1	Equality of two fuzzy sets	Fuzzy set A is considered equal to a fuzzy set BIF AND ONLY IF (iff) mA (x) = mB (x)
P2	Inclusion of one set into another set	Fuzzy set AÌ X is included                   ~ in (is a subset of) another fuzzy set, BÌX                                                         ~IF AND ONLY IF (iff) mA(x) £mB(x) " xΠX
		Example: Consider X = {1, 2, 3}and A = 0.3/1 + 0.5/2 + 1/3 B = 0.5/1 + 0.55/2 + 1/3 Then A is a subset of B
P3	Cardinality of a fuzzy set	Cardinality of a non-fuzzy set, Z, is the number of elements in Z. BUT the cardinality of a fuzzy set A, the so-called SIGMA COUNT, expressed as a SUM of the values of the membership function of A, mA(x)Card A = mA(x1) + ….. +mA(xn)     n = SmA(x1)    i=1
		EXAMPLE Card A = 1.8 Card B = 1.85
P4	An empty fuzzy set	A fuzzy set A is empty IF AND ONLY IF mA(x) = 0, " xΠX
P5	a -cuts or a -level sets of a fuzzy set	An a -cut or a -level set of a fuzzy set  A Ì X is an     ~ ORDINARY SET AaÌ X,                                    ~ such that Aa ={xΠX; mA(x)³a } 0£a£ 1 Decomposition A=Sa Aa 0£a£ 1
		Example: A=0.3/1 + 0.5/2 + 1/3 X = {1, 2, 3} Then A0.5 = {2, 3} A0.1 = {1, 2, 3} A1 = {3}