Foundations of FS Theory

Foundations of FS Theory - Foundations of Fuzzy Sets Theory...

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Foundations of Fuzzy Sets Theory 2.1. Basic Concepts and Properties of Fuzzy Sets 2.1.1. Basic types of fuzzy sets Definition. Fuzzy sets Let X be a classical set of objects, called the universe, whose generic elements are denoted x. Membership in a classical subset A of X is often viewed as a characteristic function μ A from X to {0,1} such that μ A (x) = A x if 0 A x if 1 where {0,1} is called a valuation set; 1 indicates membership while 0 - non-membership. If the valuation set is allowed to be in the real interval [0,1], then A is called a fuzzy set [1-6]. μ A (x) is the grade of membership of x in A ] 1 , 0 [ X : A μ . As closer the value of μ A (x) is to 1, so much x belongs to A. A is completely characterized by the set of pair. } X x )), x ( , x {( A A μ = . Example A=0.1/1+0.3/2+0.5/3+0.7/4+0.8/5+0.9/6+0.95/7+1.0/8+0.97/9+ +0.9/10+0.8/11+0.7/12+0.5/13+0.3/14+0.1/15. A is a fuzzy set. A classical version of this fuzzy set is A={1,2,3,4,5,6,7,8,9,10,11,12,13,14,15} or A=1/1+1/2+1/3+1/4+1/5+1/6+1/7+1/8+1/9+1/10+1/11+1/12+1/13+ +1/14+1/15. For graphical representation see Figure 2.1. Figure 2.1. Fuzzy set 0.5 1.0 5 10 15 20 x ) x ( A μ
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Example The representation of temperature within a range ] T , T [ 2 1 by fuzzy and crisp sets is shown in Figure 2.2a and 2.2b, respectively. In the first case we use membership function ] 1 , 0 [ ] T , T [ 2 1 for describing linguistic concepts "cold", "normal", "warm" . In the second case right - open intervals are used for describing of traditional variable by crisp sets. Figure 2.2. Representation of temperature by fuzzy (Figure 2.2a) and crisp (Figure 2.2b) sets Fuzzy sets with crisply defined membership functions are called ordinary fuzzy sets. If membership function of a fuzzy set A assigns to each element of the universal set X a closed interval of real numbers, then this type of fuzzy sets are called interval-valued fuzzy sets [ ] 1 , 0 X : A ε μ , where [] ε 01 , denotes the family of all closed intervals of real members in [ ] , . Example n n n A 1 1 1 A x / ) x , x ( x / ) x , x ( A 2 1 2 1 α α α α μ + + μ = K Membership function of interval-valued fuzzy set A is given in Figure 2.3. Figure 2.3. Interval-valued fuzzy set A μ A x () 1 α 2 α 1 x 1 x Cold Normal Warm T 2 T 1 μ A x Warm Normal Cold x T 2 b 1 a T 1 [ )[ )
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Fuzzy sets whose membership function itself is an ordinary fuzzy set are called fuzzy sets of type 2. Their membership functions are [ ] ( ) 1 , 0 F X : A μ , (2.1) where [] () F01 , is the set of all ordinary fuzzy sets that can be defined in [ ] 01 , , i.e. fuzzy power set of , . Example Fuzzy set of type 2 is shown in Figure 2.4. If x=x 1 four numbers α 1 , α 2 , α 3 , α 4 are produced, by which the ordinary fuzzy set defined with trapezoidal membership function assigned to x 1 is determined. The fuzzy sets of still higher types could be obtained recursively in similar way. If fuzzy set in X has membership function as type m-1, m>1 fuzzy set on [0,1], then that set is reffered to as fuzzy set type m.
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This note was uploaded on 10/12/2010 for the course CS dt 3007 taught by Professor Hussain during the Spring '10 term at Aligarh Muslim University.

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Foundations of FS Theory - Foundations of Fuzzy Sets Theory...

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