Intuitionistic Fuzzy Sets

Transcription

1 INT. J. IOUTOMTION, 2016, 20(S1), S1-S6 Intuitionistic Fuzzy Sets Krassimir T. tanassov* Copyright 1983, 2016 Krassimir T. tanassov Copyright 2016 Int. J. ioautomation. Reprinted with permission How to cite: tanassov K. T. Intuitionistic Fuzzy Sets, VII ITKR Session, Sofia, June 1983 (Deposed in Centr. Sci.-Techn. Library of the ulg. cad. of Sci., 1697/84) (in ulgarian). Reprinted: Int. J. ioautomation, 2016, 20(S1), S1-S6. I. Let E be an arbitrary fixed set and be its subset (denote E). In the theory, the fact that element x of set E belongs to is denoted by x.. 1 There, it is introduced the characteristic function : E [0,1], defined by 1, if ( x) = 0, if x. x * * Fuzzy subset [1] of E is each set with the form { x, ( x) x E}( = ), where * : E { a 0 a 1 & a M} is a function, determining the degree of membership of the element x to some set, about M, that is some numerical set. If M = {0, 1}, then this * function coincides with function. y, in [1], different relations and operations over fuzzy sets are introduced: (1) iff ( )( ( x) (2) = iff ( x E)( ( x) = (3) iff ( )( ( x) { } { } { x } * * * (4) = x, ( x), and ( x) = 1 ( x); * * (5) = x, ( x) x E, and ( x) = min( ( x), * * (6) =, * * (7) = ( ) ( ); (8) = ; ( x) x E, and ( x) = max( ( x), * * {{ } } {.. } * * { + + x x} (9) P ( ) = x, ( x) ( x) M & M M ; M (10). = x,, and = ( x). ( x) ; (11) + = x,, and = ( x) + ( x) ( ). ( ). * Current affiliation: ioinformatics and Mathematical Modelling Department Institute of iophysics and iomedical Engineering, ulgarian cademy of Sciences 105 cad. G. onchev Str., Sofia 1113, ulgaria, krat@bas.bg S1

2 INT. J. IOUTOMTION, 2016, 20(S1), S1-S6 ehind all these definitions, the xiom for Excluded Middle is seen, i.e., it is supposed that if for the element x is not valid that x, then x and opposite. Our next research will be directed t introducing of definitions similar to the above ones, but with lack of this xiom, i.e., we will obtain a theory from intuitionistic type. Let us suppose that there is a situation, in which besides the cases x and x to be possible a third case when we cannot determine which of the two cases is valid. Therefore, (x) + (x) 1. When there is =, we obtain the ordinary fuzzy sets, in the opposite case, the new objects must be used the intuitionistic fuzzy sets. Let us discuss one example. Let E = {0,1,2,...,10}, M = {0,1} R, = {n n E& the equation x 3 y 2 = n has more than 2 solutions}. For example, (0) = 1, (1) = 0, but (7) cannot be determined, because it is not known whether the equation x 3 y 2 = n has other solutions, except (2,1) and (32,181). Let us put ν (x) = (x). Therefore, 0 (x),ν (x) 1, 0 (x) + ν (x) 1. The intuitionistic fuzzy set has the form: { x, (x),ν (x) x E}. For it, (1)-(3), (7) and (8) are not changed and the rest equalities obtain the form: (4) = { x,ν (x), (x) x E}, and (x) = ν (x),ν (x) = (x); (5) = { x, (x),ν (x) x E}, and (x) = min( (x), (x)),ν (x) = max( (x), (x)); (6) = { x, (x),ν (x) x E}, and (x) = max( (x), (x)),ν (x) = min( (x), (x)); (9) P M (E) = {{ x, (x),ν (x) (x) M,ν (x) M } x E, M M,M M}; (10). = { x,.,ν. x E},, and. = (x). (x),ν. = ν + = ν (x) + ν (x) ν (x).ν (x); (11) + = { x, +,ν + x E},, and + = (x) + (x) (x). (x),ν + = ν (x).ν (x). Similarly to [1], the following assertions are valid. Theorem 1: Operations and are commutative, associative, distributive between them from left and right sides, idempotent and satisfying De Morgan s Laws. S2

3 INT. J. IOUTOMTION, 2016, 20(S1), S1-S6 Theorem 2: Operations. and + are commutative, associative and satisfying De Morgan s Laws. Theorem 3: Operations. and + are distributive from left and right sides about operations and. II. In fuzzy sets theory, the Hemming s distance between two sets is defined by D(,) = (x) (x) = R (x) (x) dx, for M = R, and Euclid s distance is defined by E (,) = ( (x) (x))2 = R ( (x) (x))2 dx, for M = R. nalogues of these definitions, here have the forms: D (,) = 2 1 ( (x) (x) + ν (x) ν (x) ) = 1 2 R and Euclid s distance is defined by ( (x) (x) + ν (x) ν (x) )dx, for M = R, E (,) = 1 2 (( (x) (x))2 + (ν (x) ν (x))2 ) = 1 2 R (( (x) (x))2 + (ν (x) ν (x))2 )dx, for M = R. III. We will call that set α,β is a set of level (α,β ) if α,β = {x x E& (x) α&ν (x) β}. Theorem 4: a) β (0 β 1) : α 2 α 1 α2,β α1,β, b) α(0 α 1) : β 1 β 2 α,β1 α,β2. S3

4 INT. J. IOUTOMTION, 2016, 20(S1), S1-S6 IV. Now, we define two operators, that transform each intuitionistic fuzzy set to a fuzzy set. They are analogous of operators necessity and possibility, that are defined in the modal logic. For each set E: = { x, (x),ν (x) x E} = { x, (x),1 (x) x E}, = { x, (x),ν (x) x E} = { x,1 ν (x),ν (x) x E}. The connection between these two operators is given by the following: Theorem 5: For each intuitionistic fuzzy set : a) =, b) =. Really, = { x,ν (x), (x) x E} = { x,ν (x),1 ν (x) x E} = { x,1 ν (x),ν (x) x E} =. = { x,ν (x), (x) x E} = { x,1 (x), (x) x E} = { x, (x),1 (x) x E} =. Therefore, the two operators are defined correctly. The connection between them is given by: Theorem 6: For each intuitionistic fuzzy set :. References 1. Kaufmann. (1977). Introduction a la Theorie des Sour-ensembles Flous, Paris, Masson. S4

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