Functions Defined on Fuzzy Real Numbers According to Zadeh s Extension

Intenational Mathematical Foum, 3, 2008, no. 16, 763-776 Functions Defined on Fuzzy Real Numbes Accoding to Zadeh s Extension Oma A. AbuAaqob, Nabil T. Shawagfeh and Oma A. AbuGhneim 1 Mathematics Depatment, Univesity of Jodan, Amman 11942, Jodan Abstact In this pape we pesent two theoems that ely on the Zadeh s extension pinciple. These two theoems can be used to define a cisp function on a given fuzzy eal numbe. And this will poduce a new fuzzy eal numbe. Using this, we can define some special fuzzy numbes such as: squae oot, natual logaithm, logaithm,...etc. Mathematics Subject Classification: 26E50; 46S40 Keywods: Fuzzy Numbe; -cut Repesentation; Absolute Value; Squae Root; Logaithm; Exponential; Isometic 1 Intoduction In the eal wold, the data sometimes cannot be ecoded o collected pecisely. Fo instance, the wate level of a ive cannot be measued in an exact way because of the fluctuation and the tempeatue in a oom also can not be measued pecisely because of a simila eason. Theefoe fuzzy numbes povide fomalized tools to deal with non-pecise quantities possessing nonandom impecision o vagueness. Thus a moe appopiate way to descibe the wate level is to say that the wate level is aound 25 metes. The phase aound 25 metes can be egaded as a fuzzy numbe 25, which is usually denoted by the capital lette A. Zadeh in [6] intoduced the concept of fuzzy set and its applications. In [1] Dubois and Pade intoduced the notion of fuzzy eal numbes and established some of thei basic popeties. Goetschel and Voxman in [12] intoduced new equivalent definition of fuzzy numbes using the paametic epesentation (cut epesentation). In [7] Zadeh poposed a so called Zadeh s extension pincipal, which played an impotant ule in the fuzzy set theoy and its applications. This extension has been studied and applied by many authos 1 o.abughneim@ju.edu.jo

764 O. A. AbuAaqob, N. T. Shawagfeh and O. A. AbuGhneim including: Nguyen [5] in analyzing this extension, Baos [9] in the analysis of the continuity of this extension, Roman [11] in the analysis of discete fuzzy dynamical systems and continuity of such extension, and Belohlavek [14] in the study of similaity. The stuctue of this pape is as follows. In section 2 we pesent the peviously obtained esults that will be used in this pape. In section 3 we addess the extension pinciple and some useful theoems. In section 4 we pove the main theoems. In section 5, we use these theoems to define some special fuzzy numbes such as the squae oot, exponential, and logaithm of a fuzzy numbe. 2 Peliminay Notes In this section, we will intoduce the basic notion of fuzzy eal numbes. Thoughout this pape the following notation will be used: R is the set of eal numbes. R I is the set of closed bounded intevals of R; R I : {I I [a, b] a b, a, b R}. A fuzzy set on R is a function fom R to [0, 1]. We will denote the set of fuzzy sets on R by F (R). Definition 1. [13] A fuzzy set A on R is a fuzzy numbe if the following conditions hold: 1. A is uppe semicontinuous. 2. Thee exist thee intevals [a, b], [b, c], and [c, d] such that A is inceasing on [a, b], A 1on [b, c], A is deceasing on [c, d], and A 0on R [a, d]. Let A be a fuzzy numbe then fo all [0, 1] the cut is defined as follows: { {x x R,A(x) } if (0, 1] cut (A) {x x R,A(x) > 0} if 0 [ A (),A + () ], whee A () min ( cut (A)) and A + () max ( cut (A)), usually we denote cut (A) bya. We give an altenative definition of a fuzzy numbe. This definition have been given by Goetschel and Voxman, see [12]. This definition elays on the cut epesentation. Definition 2. [12] A fuzzy numbe A is completely detemined by a pai A : (A,A + ) of functions A ± :[0, 1] R. These two functions define the endpoints of A and satisfy the following conditions:

Functions defined on fuzzy eal numbes 765 1. A : A () R is a bounded inceasing left continuous function (0, 1] and ight continuous fo 0. 2. A + : A + () R is a bounded deceasing left continuous function (0, 1] and ight continuous fo 0. 3. A () A + (), [0, 1]. Lemma 1. [12] Suppose that A : R [0, 1] is a fuzzy set, then A is a fuzzy numbe if and only if the following conditions hold 1. The cut: A is a closed bounded inteval fo each [0, 1]. 2. The 1 cut: A 1 φ. Moeove the membeship function A is defined by if x A o x A +, (0, 1) A (x) 1 if x A 1 0 if x / A 0. (1) We denote the set of fuzzy numbes by R F. A fuzzy numbe A is said to be positive (nonnegative) if A (0) > 0 (A (0) 0), and negative (nonpositive) if A + (0) < 0(A + (0) 0). We denote the set of positive and nonnegative fuzzy numbes by R + F and R + F {χ 0} espectively, whee χ 0 : R {0, 1} defined by χ 0 (x) 1ifx 0 and χ 0 (x) 0ifx 0. On R F, we define a patial ode by: let A, B R F, then A B iff A B (In othe wod A () B () and A + () B + (), [0, 1]). The distance between two fuzzy numbes is defined by the distance function [10]: defined by D : R F R F [0, ) D (A, B) sup [0,1] { max { A () B (), A + () B + () }}. Now we pesent some impotant esults fom functional analysis. We will use these esults in ou wok. Theoem 1. [15] (Tietze s Extension Theoem) X is nomal if and only if wheneve C is a closed subset of X and f : C R is continuous, thee is an extension of f to all of X; i.e. thee is a continuous map g : X R such that g C f.

766 O. A. AbuAaqob, N. T. Shawagfeh and O. A. AbuGhneim Definition 3. [2] (Isometic Mapping, Isometic Spaces) Let X (X, D X ) and Y (Y,D Y ) be metic spaces. Then: a. A mapping T fom X to Y is said to be isometic o an isomety if T peseves distance, that is, if fo all x, y X, D Y (T (x),t (y)) D X (x, y), whee T (x) and T (y) ae the images of x and y, espectively. b. The space X is said to be isometic to the space Y if thee exists a bijective isomety fom X to Y. The two space X and Y ae then called isometic spaces. Remak 1. A homeomophism is a continuous bijective mapping T : X Y whose invese is continuous. If thee is a homeomophism fom a metic X to Y then we say X and Y ae homeomophic. If X and Y ae isometic then they ae homeomophic, see [2]. 3 Extension Pinciple and Aithmetic Opeations In [7], Zadeh poposed a so called extension pinciple which became an impotant tool in fuzzy set theoy and its applications. Next we explain this pinciple: Let U, V, W R and f be a cisp function f : U V W. Assume A and B ae two fuzzy subsets on U and V espectively. By the extension pinciple, we can use the cisp function f to induce a fuzzy-valued function F : F (U) F(V ) F(W ). That is to say, F (A, B) is a fuzzy subset of W with membeship function { supf(x,y)z {min {A (x),b(y)}}, f F (A, B)(z) 1 (z) φ 0, f 1 (z) φ, (2) whee f 1 (z) {(x, y) U V : f (x, y) z W }. Such a function F is called a fuzzy function induced by the extension pinciple. Suppose that f : R R R is a given cisp function. Then the next theoem put some estictions on f to poduce a well-defined function F fom R F R F to R F.

Functions defined on fuzzy eal numbes 767 Theoem 2. [5] Let f : R R R be a continuous function, then F is a well-defined function fom R F R F to R F with cut fo evey A, B R F and [0, 1]. (F (A, B)) f (A,B ), Usually we denote (F (A, B)) by [F (A, B)(),F + (A, B)()]. Theoem 3. [11] Let f : R R R be a function. conditions ae equivalent Then the following i. f is continuous. ii. F : R F R F R F is continuous with espect to the metic D. The basic aithmetic opeations between two closed bounded intevals ae defined by A B {a b a A, b B}, (3) whee {+,,, } and in the division case we equie that 0 / B. Fo {+,,, } and A, B R I, then C : A B is a closed bounded inteval and the endpoints of C ae calculated as follows, see [4]. A + B [ A + B,A + + B +], A B [ A B +,A + B ], A B [min X, max X], whee X { A B,A B +,A + B,A + B +}, A B A [ 1/B +, 1/B ], povided that 0 / B, whee A [A,A + ] and B [B,B + ]. Aithmetic opeations on fuzzy numbes ae defined in tems of the wellestablished aithmetic opeations on closed bounded intevals of eal numbes, and this is by employing the cut epesentation. Let A and B denote fuzzy numbes, and let {+,,, } denotes any of the fou basic aithmetic opeations. Each one of these opeations define a continuous functions. Hence using Theoem (2) these opeations will define a fuzzy numbe as follows (A B) {x y x A,y B }, whee A and B ae the cuts of the fuzzy numbes A and B espectively. When the opeation is division it is equied that 0 / B 0.

768 O. A. AbuAaqob, N. T. Shawagfeh and O. A. AbuGhneim If A, B R F, then A and B ae closed bounded intevals fo evey [0, 1] and hence using Equation (3) and Theoem (2), we get (A B) [min x y x A,y B, max x y x A,y B ] A B, whee the membeship function (A B)(x) is given by Equation (1). Let A, B R F and λ R, then the sum A+B and the scala multiplication λa is then given by and (A + B) A + B (λa) λa [ min { λa (),λa + () }, max { λa (),λa + () }] espectively fo evey [0, 1], whee (A + B)(x) and (λa)(x) ae given by Equation (1). 4 Main Results Let X be a subset of R, though this section we will use the following symbols: X F : {A A : X [0, 1] A R I, [0, 1] and A 1 φ} and R XF {A R F A 0 X}, whee A 0 and A 1 ae the 0 cut and 1 cut of A espectively. We define a metic D X on R XF by: D X : R XF R XF [0, ) with D X (A, B) D (A, B). Let U and V be closed intevals in R, then fo evey [0, 1], we have the following esults. Theoem 4. Let f : U V R be a continuous function, then f can be extend to a well-defined continuous function with cut F : R UF R V F R F (4) (F (A, B)) f (A,B ). Poof. Since U V is a closed subset of R R and R R is a nomal space (as R R is a metic space), then using Theoem (1), thee exist a continuous function g : R R R such that g U V f.

Functions defined on fuzzy eal numbes 769 Using Theoem (2), we get, G : R F R F R F is a well-defined continuous function and (G (A, B)) g (A,B ), whee G is the function induced by Equation (2). Let F be the estiction of G on R UF R V F (as R UF R V F R F R F ), i.e. F G R U F R VF. We get F : R UF R V F R F is a well-defined continuous function. If (A, B) R UF R V F, then the cut of (F (A, B)) is given by (F (A, B)) (G (A, B)) g (A,B )f (A,B ). The last equality holds because A U and B V. Coollay 1. The ange of F is a subset of R f(u,v )F and hence F : R UF R V F R f(u,v )F is a well-defined continuous function. Poof. Want to show that the ange of F is a subset of R f(u,v )F. Since fo each (X, Y ) R UF R V F, we have F (X, Y ) R F, and (F (X, Y )) 0 f (X 0,Y 0 ) f (U, V ). Thus F (X, Y ) R f(u,v )F o Range (F ) R f(u,v )F. In the next theoem the fuzzy numbes will be esticted on subsets U and V of R and hence the domain of the extended function will be U F V F. Let A U and B V be the fuzzy numbes that have been esticted on U and V espectively. Theoem 5. Let f : U V R be a continuous function, then 1. The spaces R UF R V F and U F V F ae isometic. 2. f can be extend to a well-defined continuous function K : U F V F R F such that K ( A U,B V ) F (A, B), Whee F is given by Equation (4) with cut ( K ( A U,B V )) f (A,B ). Poof. Define a mapping H on R UF R V F by then we have: H : R UF R V F U F V F such that H (A, B) ( A U,B V ), i. H is well-defined as: if (A, B) (C, D), then ( A U,B V ) ( C U,D V ) and hence H (A, B) H (C, D).

770 O. A. AbuAaqob, N. T. Shawagfeh and O. A. AbuGhneim ii. H is one to one as: if (A, B), (C, D) R UF R V F such that H (A, B) H (C, D), then ( ) ( ) (( ) A U,B V C U,D V. This gives A U, ( ) B U ) (( C U ), ( ) ) D U and hence (A,B )(C,D ), [0, 1]. Thus we get (A, B) (C, D). iii. H is onto as: fo all Y (Y 1,Y 2 ) U F V F, define X 1 : R [0, 1] by: { Y1 (x), x U X 1 (x) 0, x R U. Similaly define X 1 : R [0, 1] by: { Y2 (x), x V X 2 (x) 0, x R V, then X 1 R UF and X 2 R V F with X 1 U Y 1 and X 2 U Y 2. Hence if X (X 1,X 2 ), then H (X) H (X 1,X 2 ) ( ) X 1 U,X 2 V (Y1,Y 2 ) Y. iv. We will show that H peseves the distance: fo all (A, B), (C, D) R UF R V F, we define a metic D on R UF R V F by with D :(R UF R V F ) (R UF R V F ) [0, ) D (X, Y ) max {D (A, C),D(B,D)}, whee X (A, B) and Y (C, D). So we get D (H (A, B),H(C, D)) D (( ) ( )) A U,B V, C U,D V max { D ( ) U A U,C ( )} U,D V B V,D V max { D ( ) ( )} A U,C U,D B V,D V max {D (A, C),D(B,D)} D ((A, B), (C, D)). The equality befoe the last one holds because X ( X U fo each ) [0, 1] and X R UF o X R V F, whee X {A, B, C, D}. Thus R UF R V F U F V F (Isometic spaces), which poves pat (1). Now we will show pat (2). Using Remak (1), H 1 : U F V F R UF R V F is a continuous function and using Theoem (4) F : R UF R V F R F is also a continuous function. If K F H 1, we get K : U F V F R F,

Functions defined on fuzzy eal numbes 771 is a well-defined continuous function. Let (A, B) R UF R V F, then K ( ) A U,B V ( )( ) ( ( F H 1 A U,B ( ))) V F H 1 A U,B V F (A, B) with cut ( ( )) K A U,B V f (A,B ). In the same way as in Theoem (5), we will get a function J : U F V F f (U, V ) F defined by K ( ) A U,B V (F (A, B)) f (U,V ). And this function will be a well-defined continuous function. We pesent that in the following coollay. Coollay 2. J : U F V F f (U, V ) F defined by: J ( ) A U,B V (F (A, B)) f (U,V ) is a well-defined continuous function. Poof. Define L : R f(u,v )F f(u,v )F with L (A) A f (U,V ). As in the poof of the last theoem we get L is a bijective and continuous function. Hence R f(u,v )F f (U, V ) F (Isometic spaces). Define J L F H 1, then J : U F V F f (U, V ) F is a well-define continuous function and this is because F : R UF R V F R f(u,v )F and H : R UF R V F U F V F ae a well-defined continuous functions. Moeove J ( ) A U,B ( ) ( ( V L F H 1 A U,B ( ))) V L F H 1 A U,B V L (F (A, B))(F (A, B)) f (U,V ). Let A R F and f : U R be a continuous function on A 0 U. If f is an inceasing function then (F (A)) [f (A ()),f(a + ())]. If f is a deceasing function then (F (A)) [f (A + ()),f(a ())], and if f is not a monotone function then (F (A)) [min f (x) x A, max f (x) x A ]. Example 1. Define a function f : U [ɛ, ) R, ɛ>0 such that f (x) 1/x, let X R UF {A R F A 0 U}, then accoding to Zadeh s extension pinciple, we can induce a well-defined continuous function F : R UF R F with cut (F (X)) f (X ), o a well-defined continuous function K : U F R F such that K ( ) ( ( X U F (X), with cut K X U )) f (X ). Thus if X [X (),X + ()], then (F (X)) [1/X + (), 1/X ()]. In paticula if A is a fuzzy numbe with A [ +1, 3 ], then (1/A) [1/ (3 ), 1/ ( + 1)]. Fom peceding theoems, one can define the squae oot, natual logaithm, logaithm,...etc. fo a given fuzzy numbe. In the next section we will do that

772 O. A. AbuAaqob, N. T. Shawagfeh and O. A. AbuGhneim 5 Applications 5.1 The squae oot of a fuzzy numbe Definition 4. The absolute value of a fuzzy numbe X R F is a function F : R F R F denoted by F (X) : X with cut ( X ) { x x X }. Fom the inteval analysis [3], we know that if I [I,I + ], then I [max (I, I +, 0), max ( I,I + )], thus the cut of A is given by ( A ) [ max ( A (), A + (), 0 ), max ( A (),A + () )] and hence we get A if A 0 ( A ) A if A 0 [0, max ( A (),A + ())] if 0 (A (0),A + (0)). Since f (x) x is a continuous function on R, we get F (X) X is a continuous function on R F. Definition 5. The squae oot of a fuzzy numbe X R F is a function F : R UF R F denoted by F (X) : ( X ) X with cut { x x A }, whee U [0, ). Since f (x) x is a continuous function on [0, ), we get F (X) X is ( a continuous function on R UF. Because f is inceasing on [0, ), we have X ) [ X (), X ()] +. ( Example 2. Let A R F with A [, 2 ] fo each [0, 1]. Then A ) [ ], 2. Theoem 6. Let A R F, then we have 1. 2. n An A if n is an even positive intege. n An A if n is an odd positive intege. Poof. Let A [A (),A + ()]. Conside f ( : R ) R such that f (x) n xn x, then fo each [0, 1], we have n A n { } n x n x A { x x A } ( A ). Hence n A n A. Fo the second pat, the poof is simila by taking f (x) n x n x.

Functions defined on fuzzy eal numbes 773 Note that A 2 is not necessaily equal to A. To see this conside A ( R F with A [ 1, 1 ] fo each [0, 1]. Then A 2) [0, 1 ] [ 1, 1 ]. ( A 2 A only if 0 / (A (0),A + (0))). Now, we give some popeties of the squae oot of a fuzzy numbe. Obseve that these popeties ae simila to the ones fo the squae oot of eal numbes. Theoem 7. Let A, B R + F {χ 0}, then we have 1. A + B A + B. 2. AB A B. 3. A/B A/ B, povided that 0 / B 0. Poof. We pove only the fist pat. The poof of the othe pats is simila. Let A [A (),A + ()] and B [B (),B + ()], then we have (A + B) [A ()+B (),A + ()+B + ()]. Conside f :[0, ) R such that f (x) x, then fo each [0, 1], we have ( A + B ) [ ( A ) ( A ) ] + + B (), + B () [ A ()+B (), A + ()+B ()] + [ A ()+ B (), A + ()+ ] B + () [ ( ) ( ) ( ) + ( ) ] + A ()+ B (), A ()+ B () ( A ) ( ) + B ( ) A + B. Hence A + B A + B. 5.2 The Exponential and logaithm of a fuzzy numbe Definition 6. The exponential of a fuzzy numbe X R F is a function F : R F R F denoted by F (X) : exp X with cut (exp X) {exp x x X }. Since f (x) exp x is a continuous function on R, we get F (X) exp X is a continuous function on R F. Because f is inceasing on R, we get (exp X) [exp X (), exp X + ()]. Definition 7. The natual logaithm of a fuzzy numbe X R F is a function F : R + F R F denoted by F (X) :lnx with cut (ln X) {ln x x X }.

774 O. A. AbuAaqob, N. T. Shawagfeh and O. A. AbuGhneim Since f (x) lnx is a continuous function on [ɛ, ), ɛ>0, we get F (X) ln X is a continuous function on R + F. Because f is inceasing on [ɛ, ), ɛ>0, we have (ln X) [ln X (), ln X + ()]. Example 3. Let A R F with A [ +1, 3 ] fo each [0, 1]. Then (ln A) [ln ( +1), ln (3 )]. Next, we pesent some popeties fo the natual logaithm of a fuzzy numbe. These popeties ae simila to the ones fo eal numbes Theoem 8. Fo any positive fuzzy numbe A and B ( A, B R + F) and a ational numbe α, we have 1. ln 1 0. 2. ln AB lna +lnb. 3. ln A/B lna ln B. 4. ln A α α ln A. Poof. Let A [A (),A + ()] and B [B (),B + ()], then (AB) [A () B (),A + () B + ()] and (A/B) [A () /B + (),A + () /B ()]. Conside f :(0, ) R with f (x) lnx, then fo each [0, 1], we have 1 [1, 1] and hence [ ] (ln 1) min ln x, max ln x [ln 1, ln 1] [0, 0]0, x 1 x 1 Thus we get ln 1 0. Fo pat (2), we have (ln AB) [ (ln AB) (), (ln AB) + () ] [ ln ( A () B () ), ln ( A + () B + () )] [ ln A ()+lnb (), ln A + ()+lnb + () ] [ (ln A) () + (ln B) (), (ln A) + () + (ln B) + () ] (ln A +lnb). Hence ln AB lna +lnb. Fo pat (3), we have (ln A/B) [ (ln A/B) (), (ln A/B) + () ] [ ln ( A () /B + () ), ln ( A + () /B () )] [ ln A () ln B + (), ln A + () ln B () ] [ (ln A) () (ln B) + (), (ln A) + () (ln B) () ] (ln A ln B).

Functions defined on fuzzy eal numbes 775 Hence ln A/B lna ln B. Fo the last pat, Since f (x) lnx α α ln x, then if α>0 we have (ln A α ) (α ln A) [ (α ln A) (), (α ln A) + () ] [ α ln ( A () ),αln ( A + () )] α [ ln ( A () ), ln ( A + () )] α [ (ln A) (), (ln A) + () ] α (ln A). Hence ln A α α ln A. The poof is simila if α<0. Theoem 9. Let A R F and B R + F, then we have 1. ln (exp A) A. 2. exp (ln B) B. Poof. Let A [A (),A + ()]. Conside f : R + R with f (x) exp (ln x). Then fo each [0, 1], we have (exp (ln A)) [ (exp (ln A)) (), (exp (ln A)) + () ] [ exp ln A (), exp ln A + () ] [ A (),A + () ] A. Hence exp (ln A) A. The second pat is simila Remak 2. Note that all the theoems that we stated in the pevious subsection ae also hold fo the logaithm to any base of a fuzzy numbe. In this case F : R + F R F denoted by F (X) : log β X with cut ( log β X ) { logβ x x A } and β R +. Refeences [1] D. Dubois and H. Pade, Opeations on fuzzy numbes, Intenational Jounal of Systems Science, 9 (1978), 613-626. [2] E. Keyszig, Intoductoy Functional Analysis with Applications, John Wiley and Sons, New Yok, 1989. [3] G. Alefeldt, D. Claudio, The Basic Popeties of Inteval Aithmetic; its Softwae Realizations and Some Applications, Computes and Stuctues, 67 (1998), 3-8. [4] G. Alefeld, J. Hezbege. Intoduction to Inteval Computations. Academic Pess, New Yok, 1983.

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