Geometrical Interpretation of the Elements of the Intuitionistic Fuzzy Objects

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1 Geometrical Interpretation of the Elements of the Intuitionistic Fuzzy Objects Krassimir T. Atanassov* Copyright 1989, 2016 Krassimir T. Atanassov Copyright 2016 Int. J. Bioautomation. Reprinted with permission How to cite: Atanassov K. T. Geometrical Interpretation of the Elements of the Intuitionistic Fuzzy Objects, Mathematical Foundations of Artificial Intelligence Seminar, Sofia, 1989, Preprint IM-MFAIS Reprinted: Int J Bioautomation, 2016, 20(S1), S27-S42. The concepts intuitionistic fuzzy set (IFS), intuitionistic fuzzy propositional calculus (IFPC), intuitionistic fuzzy modal logic (IFML) has been investigated yet only as logical and algebraic objects (c.f. [1-6]). Here we shall give two geometrical interpretations of the elements of IFSs, IFPC and IFML. * Let the universe E be given and let the figure F in the Euclidian plane with Cartesian coordinates be given (see Fig. 1): F = {p (p = a, b ) & (a, b 0) & (a + b 1)}. Let A E be a fixed set. Then we can construct a function f A from E to F, that if x E, then f A (x) = p = a, b F, 0 a + b 1. These coordinates are such that: a = µ A (x), b = A (x). Therefore the function f A is a surjection, because every two different elements of F can be images of elements of E about f A. The opposite is not valid. For example, if x, y E but x y and µ A (x) = µ A (y), A (x) = A (y) about the set A E, then f A (x) = f A (y). Obviously, f A will assign the element with coordinates 1, 0 to the element x E for which: µ A (x) = 1, A (x) = 0, and f A will assign the element with coordinates 0, 1 to the element x E for which: µ A (x) = 0, A (x) = 1. * Current affiliation: Bioinformatics and Mathematical Modelling Department Institute of Biophysics and Biomedical Engineering, Bulgarian Academy of Sciences 105 Acad. G. Bonchev Str., Sofia 1113, Bulgaria, krat@bas.bg S27

2 The geometrical interpretation of the operations and operators over the elements of the IFS, IFPC and IFML is important for the fact, that they become visualizable. 0,1 g A E F 0,0 Fig. 1 1,0 If x, y E, then to x & y f A will assign a point f A (x & y) F with coordinates min(µ A (x), µ A (y)), max( A (x), A (y)). There exist three geometrical cases (see Fig. 2 a-c) one general case (Fig. 2a) and two particular cases (Fig. 2b and 2c). f A (p&q) f A (p&q) f A (p&q) (a) (b) (c) Fig. 2 If x, y E, then to x y f A will assign a point f A (x y) F with coordinates max(µ A (x), µ A (y)), min( A (x), A (y)). There exist also three geometrical cases as above one general case (Fig. 3) and two particular cases. Fig. 3 S28

3 More complex is the case of the element x y (c.f. [4]) because for it there exist at least two different versions of a definition. In the first case, f A assigns to the element x y E the element f A (x y) F with coordinates 1 (1 µ A (y)).sg(µ A (x)) µ A (y)), A (y).sg(µ A (x) µ A (y)).sg( A (y) A (x)) (Fig. 4a). In contrast to the case for &, here the locations of the points f A (x) and f A (y) are important for the form of the location of the point f A (x y). The locations in Fig. 4a and 4b are general and in Fig. 4c a special case; in Fig. 4d and 4e are general and in Fig. 4f a special case; in Fig. 4g and 4h are particular cases of these from Fig. 4a and 4b. In the second case the point f A (x y) has coordinates max(µ A (y), A (x)), min( A (x), µ A (y)) and the locations of the points f A (x) and f A (y) are important for the form of the location of the point f A (x y) also see Fig. 5a-h. (a) (b) (c) (d) S29

4 (e) (f) (g) Fig. 4 (h) (p) µ(p) µ(q) (q) (a) (q) µ(p) (p) µ(q) (b) S30

5 µ(p) (q) (p)= (q) µ(q) µ(q) (c) (p) µ(p) (d) (p) µ(q) (q) µ(q) (p)= (q) µ(p) µ(p) (e) (f) S31

6 (p) (q) µ(p) = µ(q) µ(p) = µ(q) (q) (g) Fig. 5 (p) (h) Below we shall give one by one the geometrical interpretations of the different operators as follows: the operator has the form from Fig. 6; the operator has the form from Fig. 7; Fig. 6 Fig. 7 the operator D assigns to the proposition p a point on the segment between the points p, p and p depending on the value of the argument [0, 1] (see Fig. 8); the operator F, assigns to the proposition p a point on the triangle with vertices p, p and p depending on the value of the arguments, [0, 1] for which + 1 (see Fig. 9); S32

7 f A (D α (p)) f A (F α,β (p)) β α α Fig. 8 Fig. 9 the operator G, assigns to the proposition p a point on the rectangle with vertices p, pr 1 p, pr 2 p and O, where pr i p is the i-th projection (i = 1, 2) of the point p, depending on the value of the arguments, [0, 1] (see Fig. 10); β f A (G α,β (p)) α Fig. 10 the operator H, assigns to the proposition p a point on the rectangle with vertices pr 2 p, p, p, and pr 2 p, depending on the value of the arguments, [0, 1] (see Fig. 11); the operator H, assigns to the proposition p a point on the figure with vertices pr 2 p, p, p, and point 0, 1, depending on the value of the arguments, [0, 1] (see Fig. 12); S33

8 f A (H α,β (p)) Fig. 11 Fig. 12 the operator J, assigns to the proposition p a point on the rectangle with vertices pr 1 p, p, p, and pr 1 p, depending on the value of the arguments, [0, 1] (see Fig. 13); the operator J, assigns to the proposition p a point on the rectangle with vertices pr 1 p, p, p, and point 1, 0 depending on the value of the arguments, [0, 1] (see Fig. 14). f A (J α,β (p)) Fig. 13 Fig. 14 The above interpretations are the same for the elements of IFPC and IFML as for the elements of IFS. The following two interpretations are related only to the elements of IFSs. If A, B are two IFSs over E, then function f A+B will assign a point f A+B (x) F with coordinates µ A (x) + µ B (x) µ A (x).µ B (x), A (x). B (x). There exists only one geometrical interpretation of this operation (see Fig. 15a). On the Fig. 15b and 15c are shown the constructions of µ A (x).µ A (y), A (x). A (y), respectively. S34

9 f B (x) B (x) A (x) A (x). B (x) f A (x) f A+B (x) µ A (x) µ B (x) µ A (x) + µ B (x) µ A (x).µ B (x) (a) µ A (x).µ B (x) µ A (x) (b) 1 µ B (x) A (x). B (x) A (x) (c) Fig B (x) If A, B are two IFSs over E, then function f A.B will assign to x E a point f A.B (x) F with coordinates µ A (x).µ B (x), A (x) + B (x) A (x). B (x). Here also there exists only one geometrical interpretation of the operation (see Fig. 16; the constructions of µ A (x).µ A (y) and A (x). A (y) are as above). S35

10 A (x) + B (x) A (x). B (x) B (x) A (x) f A.B (x) f A (x) f B (x) µ A (x) µ A (x).µ B (x) µ B (x) Fig. 16 The inequality 0 a + b 1 between the coordinates of the point F can change to the inequation 0 a 2 + b 2 1. This fact corresponds to the following modification of the IFS, which can be constructed: for every element x E, the values of the degree of membership and the degree of nonmembership satisfy the inequation 0 µ A (x) 2 + A (x) 2 1 about some set A E. Obviously, for all real numbers a, b [0; 1], if 0 a + b 1, then 0 a 2 + b 2 1. Hence for the new generated IFS, the above defined operations, relations and operators are valid. Below we shall show the geometrical interpretation of the operators over the second type IFS, because the geometrical interpretation of the operations over them is almost identical with the above ones and the different is only to the form of the figure F. The new form is shown in Fig. 17. The new interpretation function is noted with g A, and g A : E F. S36

11 0,1 g A E F 0,0 Now the operators are interpreted as: the operator : 1,0 Fig. 17 g A ( p) g A (p) the operator : Fig. 18 g A (p) g A ( p) Fig. 19 S37

12 the operator D α : g A ( p) g A (D α (p)) g A ( p) g A (p) the operator F α,β : Fig. 20 g A ( p) g A (F α,β (p)) g A ( p) g A (p) Fig. 21 The operators G α,β, H α,β and J α,β have interpretations as above with exactness to the form of the figure F. The remaining operators are interpreted as: the operator H, : g A ( p) g H p ( ( )) A, g A (p) Fig. 22 S38

13 the operator J, : g A (p) g A ( p) g J p ( ( )) A, Fig. 23 These interpretations give an illustrative idea of the elements of IFS, IFPC and FMC and the operations and operators over them. References 1. Atanassov K. (1986). Intuitionistic Fuzzy Sets, Fuzzy Sets and Systems, 20(1), Atanassov K. (1989). More on intuitionistic fuzzy sets, Fuzzy Sets and Systems, 33(1), Atanassov K. T. (1988, 2016). Review and New Results on Intuitionistic Fuzzy Sets, Mathematical Foundations of Artificial Intelligence Seminar, Sofia, 1988, Preprint IM- MFAIS-1-88, Reprinted: Int J Bioautomation, 20(S1), S7-S Atanassov K. T. (1988, 2016). Two Variants of Intuitionistic Fuzzy Propositional Calculus, Mathematical Foundations of Artificial Intelligence Seminar, Sofia, 1988, Preprint IM-MFAIS-5-88, Reprinted: Int J Bioautomation, 20(S1), S17-S Atanassov K. T. (1989, 2016). Four New Operators on Intuitionistic Fuzzy Sets, Mathematical Foundations of Artificial Intelligence Seminar, Sofia, 1989, Preprint IM- MFAIS-4-89, Reprinted: Int J Bioautomation, 20(S1), S55-S Atanassov K. T. (1989, 2016). Intuitionistic Fuzzy Sets, Fuzzy Sets and Systems, Mathematical Foundations of Artificial Intelligence Seminar, Sofia, 1989, Preprint IM- MFAIS-3-89, Reprinted: Int J Bioautomation, 20(S1), S43-S54. Original references as presented in Preprint IM-MFAIS Atanassov K. More on intuitionistic fuzzy sets, Fuzzy Sets and Systems (in press). 3. Atanassov K. Review and new results on intuitionistic fuzzy sets, Preprint IM-MFAIS- 1-88, Sofia, Atanassov K. Two variants of intuitionistic fuzzy propositional calculus. Preprint IM-MFAIS-5-88, Sofia, S39

14 5. Atanassov K. Four new operators on intuitionistic fuzzy sets. Preprint IM-MFAIS-4-89, Sofia, Atanassov K. Intuitionistic fuzzy sets, Fuzzy Sets and Systems. Preprint IM-MFAIS-3-89, Sofia, S40

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