Twin Hypercube for Intuitionistic Fuzzy Sets and Their Application in Medicine

International Journal of Analysis and Applications ISSN 2291-8639 Volume 15, Number 1 (2017), 31-45 http://www.etamaths.com TWIN HYPERCUBE FOR INTUITIONISTIC FUZZY SETS AND THEIR APPLICATION IN MEDICINE B. DAVVAZ1,∗, E. HASSANI SADRABADI1, JUAN J. NIETO2AND ANGELA TORRES3 Abstract. In this paper, we present a description of intuitionistic fuzzy sets as vectors in twin hypercube. Finally we verify an application of intuitionistic fuzzy sets in medicine using twin hypercube for intuitionistic fuzzy sets. 1. Introduction The concept of intuitionistic fuzzy sets was introduced by Atanassov [1]. The intuitionistic fuzzy sets have some applications in different sciences. In the papers [6,9,12,13] has been developed the concept of applications of intuitionistic fuzzy sets in medical diagnosis using from relation between intuitionistic fuzzy sets and symptoms. Also this paper is a generalization of the paper [11] to intuitionistic fuzzy sets. We present some applications of intuitionistic fuzzy sets in diagnosis. Actually by employing twin hypercube for intuitionistic fuzzy sets we determine the kind of patient. This paper is organized as follows: In first section we recall basic facts about intuitionistic fuzzy sets. In second section we present twin hypercube. In third section we describe distance and entropy of intuitionistic fuzzy sets and fourth section we present intuitionistic fuzzy segments and intuitionistic fuzzy midpoints. In the final section we study an application of intuitionistic fuzzy sets in medicine. 2. Basic facts about intuitionistic fuzzy sets After the introduction of fuzzy sets by Zadeh [16], several researches consider some generalization of fuzzy sets. As an important generalization of the notion of fuzzy sets on a non-empty set X, Atanassov introduced in [1,3], the concept of intuitionistic fuzzy sets defined on a non-empty set Xas an object having the form A={hx, µA(x), νA(x)i:x∈X},(2.1) where the functions µA:X→[0,1], νA:X→[0,1], denote the degree of membership and the degree of non-membership of each element x∈Xto the set Arespectively, and 0 ≤µA(x) + νA(x)≤1 for all x∈X. If νA(x) = 1 −µA(x) , then A is a classical fuzzy set. Such defined objects are studied by many authors and have many interesting applications not only in mathematics (see [5]). Let Aand Bbe two intuitionistic fuzzy sets in X. Then, the following expressions are defined in [1,3]. (1) A⊆Bif and only if µA(x)≤µB(x) and νA(x)≥νB(x), (2) A=Bif and only if A⊆Band B⊆A, (3) AC={hx, νA(x), µA(x)i:x∈X}, (4) A∩B={hx, min{µA(x), µB(x)},max{νA(x), νB(x)}i :x∈X}, (5) A∪B={hx, max{µA(x), µB(x)},min{νA(x), νB(x)}i :x∈X}, (6) A={hx, µA(x),1−µA(x)i:x∈X}, (7) ♦A={hx, 1−νA(x), νA(x)i:x∈X}. Received 3rd May, 2017; accepted 26th June, 2017; published 1st September, 2017. 2010 Mathematics Subject Classification. 92C50, 03E72. Key words and phrases. intuitionistic fuzzy set; Hamming distance; intuitionistic fuzzy midpoint; intuitionistic fuzzy segment; medicine. c 2017 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 31 32 DAVVAZ, HASSANI SADRABADI, NIETO AND TORRES We recall the following two examples from [7]. Example 2.1. Consider the universe X={10,100,500,1000,1200}. An intutionistic fuzzy set “Large” of Xdenoted by Land may be defined by L={h10,0.01,0.9i,h100,0.1,0.88i,h500,0.4,0.5i, h1000,0.8,0.1i,h1200,1,0i}. One may define an intuitionistic fuzzy set “Very Large” (denoted by V L) as follows: µV L(x)=(µL(x))2and νV L(x)=1−(1 −νL(x))2, for all x∈X. Thus, V L ={h10,0.0001,0.99i,h100,0.01,0.9856i,h500,0.16,0.75i, h1000,0.64,0.19i,h1200,1,0i}. Example 2.2. Consider the universe {a1, a2, a3, a4, a5, a6}. Let Aand Bbe two intuitionistic fuzzy sets of Xgiven by A={ha1,0.2,0.6i,ha2,0.3,0.7i,ha3,1,0i, ha4,0.8,0.1i,ha5,0.5,0.4i} and B={ha1,0.4,0.4i,ha2,0.5,0.2i,ha3,0.6,0.2, ha4,0.1,0.7i,ha5,0,1i}. Then, Ac={ha1,0.6,0.2i,ha2,0.7,0.3i,ha3,0,1i, ha4,0.1,0.8i,ha5,0.4,0.5i}, A∩B={ha1,0.2,0.6i,ha2,0.3,0.7i,ha3,0.6,0.2i, ha4,0.1,0.7i,ha5,0,1i}, A∪B={ha1,0.4,0.4i,ha2,0.5,0.2i,ha3,1,0i, ha4,0.8,0.1i,ha5,0.5,0.4i}, A={ha1,0.2,0.8i,ha2,0.3,0.7i,ha3,1,0i, ha4,0.8,0.2i,ha5,0.5,0.5i}, ♦B={ha1,0.6,0.4i,ha2,0.8,0.2i,ha3,0.8,0.2i, ha4,0.3,0.7i,ha5,0,1i}. For each intuitionistic fuzzy set in X, we call πA(x) = 1 −µA(x)−νA(x), the intuitionistic index of xin A. It is a hesitancy degree of xto A[2,4]. 3. Twin hypercube Similar to Kosko [10], we introduce a geometrical interpretation of intuitionistic fuzzy sets as vectors in twin hypercube. Indeed, for a given set X={x1, . . . , xn}, an intuitionistic fuzzy set Ais determined just by two mappings µA:X−→ I= [0,1] and νA:X−→ I= [0,1], and the values µA(x) and νA(x) giving the degree of membership and the degree of non-membership of each element x∈Xto the intuitionistic fuzzy set A, respectively. Thus, the set of all intutionistic fuzzy sets in Xis determined precisely by two unit hypercubes. One of them is the unit hypercube of membership degree and another one is the unit hypercube of non-membership degree. Any intuitionistic fuzzy set Adetermines a vector P∈In×In given by h(µa(x1), . . . , µA(xn),(νA(x1), . . . , νA(xn)i. Conversely, any vector Q=h(a1, . . . , an),(b1, . . . , bn)igenerates an intuitionistic fuzzy set Adefined by µA(xi) = a1and νA(xi) = b1, for i= 1, . . . , n. Cardinality of intuitionistic fuzzy sets [15]. Let Abe an intuitionistic fuzzy set in X. First, we consider the following two cardinalities of an intuitionistic fuzzy set. TWIN HYPERCUBE FOR INTUITIONISTIC FUZZY SETS AND THEIR APPLICATION IN MEDICINE 33 (1) The least cardinality of Ais equal to the so-called sigma-count (cf. [17,18]), and is called the min PCount (min-sigma-count): min XCount(A) = n X i=1 µA(xi). (2) The biggest cardinality of A, which is possible due to πA, is called the max PCount (maxsigma-count), and is equal to max XCount(A) = n X i=1 (µA(xi) + πA(xi)). Clearly, for AC, we have min PCount(Ac) = n X i=1 νA(xi), max PCount(Ac) = n X i=1 (νA(xi) + πA(xi)). Now, the cardinality of an intuitionistic fuzzy set Ais defined as the interval Card(A) = |A|=hmin XCount(A),max XCount(A)i. Therefore, we have |A|=hn X i=1 µA(xi), n X i=1 (1 −νA(xi))i.(3.1) Theorem 3.1. If Aand Bare two intuitionistic fuzzy sets in X, then |A∪B|=|A|+|B|−|A∩B|.(3.2) Proof. According to Eq. (3.1), we have |A∪B|=hn X i=1 µA∪B(xi), n X i=1 (1 −νA∪B(xi))i =hn X i=1 max{µA(xi), µB(xi)}, n X i=1 (1 −max{νA(xi), νB(xi)})i =hn X i=1 max{µA(xi), µB(xi)}, n X i=1 min{1−νA(xi),1−νB(xi)}i, |A∩B|=hn X i=1 µA∩B(xi), n X i=1 (1 −νA∩B(xi))i =hn X i=1 min{µA(xi), µB(xi)}, n X i=1 (1 −min{νA(xi), νB(xi)})i =hn X i=1 min{µA(xi), µB(xi)}, n X i=1 max{1−νA(xi),1−νB(xi)}i. Therefore, we obtain |A∪B|+|A∩B|=hn X i=1 µA(xi) + µB(xi),(1 −νA(xi)) + (1 −νB(xi))i =hn X i=1 µA(xi), n X i=1 (1 −νA(xi))i+hn X i=1 µB(xi), n X i=1 (1 −νB(xi))i =|A|+|B|.  Remark 3.1. When i= 1, for simplicity we use the symbol max Count(A)insteas of max PCount(A). 34 DAVVAZ, HASSANI SADRABADI, NIETO AND TORRES 4. Distance and entropy of intuitionistic fuzzy sets The above discussion allows us to consider the following distance between intutionistic fuzzy sets. For two intutionistic fuzzy sets Aand B, The Hamming distance [14,15] between Aand Bare given by d(A, B) = n X i=1 |µA(xi)−µB(xi)|+|νA(xi)−νB(xi)|+|πA(xi)−πB(xi)|.(4.1) Each summand in (4) is between 0 and 1. For A, B, C intuitionistic fuzzy sets, we have: (1) 0 ≤d(A, B)≤n, (2) d(A, B) = 0 if and only if A=B, (3) d(A, B) = d(B, A), (4) d(A, B)≤d(A, C) + d(C, B). A geometric interpretation of intutionistic fuzzy sets is presented in [15]. An intuitionistic fuzzy set is represented by a point in the triangle A0B0D0, where A0(1,0,0), B0(1,0,0) and D0(0,0,1). An intutionistic fuzzy set Ain Xis mapped into triangle A0B0D0in that each element of Xcorresponds to an element of A0B0D0, as an example, a point x0∈A0B0D0corresponding to x∈Xis marked the values of µA(x), νA(x) and πA(x). Let Ebe a fuzzy set defined on fuzzy sets. Then, Eis an entropy measure if it satisfies the four De Luca and Termini axioms [8]. According to a geometric representation of an intuitionistic fuzzy set [15], we have E(A) = a b,(4.2) where ais the distance (A, Anear) from Ato the nearest point Anear among A0and B0, and bis the distance (A, Afar) from Ato the fareast point Afar among A0and B0. The Eq. (4.2) describes the degree of fuzziness for a single point belonging to an intuitionistic fuzzy set. For npoints belonging to an intuitionistic fuzzy set we have E(A) = 1 n n X i=1 E(Ai).(4.3) Theorem 4.1. [15] A generalized entropy measure of an intuitionistic fuzzy set Ain Xis E(A) = 1 n n X i=1 max Count(Ai∩Ac i) max Count(Ai∪Ac i)=1 n n X i=1 n X j=1 µAi∩Ac i(xj) + πAi∩Ac i(xj) n X j=1 µAi∪Ac i(xj) + πAi∪Ac i(xj) =1 n n X i=1 n X j=1 min{µAi(xj), νAi(xj)}+ 1 −µAi∩Ac i−νAi∩Ac i n X j=1 max{µAi(xj), νAi(xj)}+ 1 −µAi∪Ac i−νAi∪Ac i =1 n n X i=1 n X j=1 min{µAi(xj), νAi(xj)}+ 1 −2 min{µAi(xj), νAi(xj)} n X j=1 max{µAi(xj), νAi(xj)}+ 1 −2 max{µAi(xj), νAi(xj)} =1 n n X i=1 n X j=1 −min{µAi(xj), νAi(xj)}+ 1 n X j=1 −max{µAi(xj), νAi(xj)}+ 1 . TWIN HYPERCUBE FOR INTUITIONISTIC FUZZY SETS AND THEIR APPLICATION IN MEDICINE 35 where Aidenotes the single element intutionistic fuzzy set corresponding to the ith element of the universe X. In other words, Aiis the ith component of A. For two intuitionistic fuzzy subsets Aand B, it is possible to define a degree of subsethood S(A, B) = |A∩B| |A|. For the midpoint M, since µM(xj) = νM(xj) = 1 2, we have E(M) = 1 n n X i=1 n X j=1 1 2 n X j=1 1 2 =1 n n X i=1 n 2 n 2 =1 n n X i=1 1 = n n= 1. Since for any intutionistic fuzzy subset A6=M, we have E(A)<1, therefore the midpoint Mis maximally intutionistic fuzzy set. Now we present the following definition. The intuitionistic fuzzy subset Cis between intuitionistic fuzzy subsets Aand B, if d(A, C) + d(C, B) = d(A, B). The metric segment between Aand Bis defined as the following: segment(A, B) = {C:d(A, C) + d(C, B) = d(A, B)}, that is, n X i=1 |µA(xi)−µC(xi)|+|µC(xi)−µB(xi)|= n X i=1 |µA(xi)−µB(xi)|, n X i=1 |νA(xi)−νC(xi)|+|νC(xi)−νB(xi)|= n X i=1 |νA(xi)−νB(xi)|. Another concept is the idea of equidistant points: equid(A, B) = {C:d(A, C) = d(C, B)}, that is, n X i=1 |µA(xi)−µC(xi)|= n X i=1 |µC(xi)−µB(xi)|, n X i=1 |νA(xi)−νC(xi)|= n X i=1 |νC(xi)−νB(xi)|. If C∈segment(A, B), and it is equidistant to Aand B, then d(A, C) = d(C, B) = 1 2d(A, B). Therefore we introduce the set of midpoints between Aand B: mid(A, B) = {C:d(A, C) = d(C, B) = 1 2d(A, B)}, that is, n X i=1 |µA(xi)−µC(xi)|= n X i=1 |µC(xi)−µB(xi)|=1 2 n X i=1 |µA(xi)−µB(xi)|, n X i=1 |νA(xi)−νC(xi)|= n X i=1 |νC(xi)−νB(xi)|=1 2 n X i=1 |νA(xi)−νB(xi)|. Obviously for any two intuitionistic fuzzy subsets Aand Bwe have, mid(A, B)⊆equid(A, B). 36 DAVVAZ, HASSANI SADRABADI, NIETO AND TORRES The canonical midpoint Aand Bis denoted by µC(xi) = 1 2(µA(xi) + µB(xi)), νC(xi) = 1 2(νA(xi) + νB(xi)), i = 1,2, . . . , n. This canonical midpoint is not the unique midpoint and there are more midpoints. Actually we show Cis a midpoint between Aand B: n X i=1 |µA(xi)−1 2(µA(xi) + µB(xi))|= n X i=1 1 2|µA(xi)−µB(xi)|, n X i=1 |νA(xi)−1 2(νA(xi) + νB(xi))|= n X i=1 1 2|νA(xi)−νB(xi)|. Therefore d(A, C) = 1 2d(A, B) and similarly d(C, B) = 1 2d(A, B), then d(A, C) = d(C, B) = 1 2d(A, B). If Ais a crisp subset of X, that is, Ais a vertex of the hypercube, then µA:X→ {0,1}, νA:X→ {0,1}, therefore d(A, M) = n X i=1 |µA(xi)−1 2|+|νA(xi)−1 2|+|πA(xi)−(1 −µM(xi)−νM(xi))| = n X i=1 1 2+1 2= n X i=1 1 = n, and d(A, Ac) = n X i=1 |µA(xi)−µAc(xi)|+|νA(xi)−νAc(xi)|+|πA(xi)−πAc(xi)| = n X i=1 |µA(xi)−νA(xi)|+|νA(xi)−µA(xi)|+|πA(xi)−(1 −νA(xi)−µA(xi))| = n X i=1 2|µA(xi)−νA(xi)|= n X i=1 2 = 2n. Similarly d(Ac, M) = n, and then d(A, M) = d(Ac, M) = 1 2d(A, Ac). Hence n X i=1 |µA(xi)−1 2|=1 2 n X i=1 |µA(xi)−µAc(xi)|=n 2, n X i=1 |νA(xi)−1 2|=1 2 n X i=1 |νA(xi)−νAc(xi)|=n 2. Since d(A, C) = d(C, B) = 1 2d(A, B), it follows that d(A, C)+d(C, B) = 1 2d(A, B) and so mid(A, B)⊂ segment(A, B). Hence, C∈segment(A, B). In the bellow example we describe the definitions Example 4.1. Let X={x1, x2},µA={0,0}, νA={1,1}, µB={1,1}, νB={0,0}. Then, we have µA+µB 2= (1 2,1 2),νA+νB 2= (1 2,1 2)∈mid(A, B). If µC= (c1, c2), νC= (c3, c4)is another midpoint between Aand Bthen: d(A, C) = c1+c2+ (1 −c3) + (1 −c4), d(A, B) = n X i=1 |µA(xi)−µB(xi)|+|νA(xi)−νB(xi)|= n X i=1 (2 + 2) = 4, d(C, B)=1−c1+ 1 −c2+c3+c4⇒c1+c2= 1, c3+c4= 1. Therefore •d(A, C) = 1 2d(A, B)⇒1−c1+ 1 −c2= 1 ⇒c1+c2= 1, TWIN HYPERCUBE FOR INTUITIONISTIC FUZZY SETS AND THEIR APPLICATION IN MEDICINE 37 •d(C, B) = 1 2d(A, B)⇒c3+c4= 1, so mid(A, B) = {C: (c1, c2),(c3, c4)|c1+c2= 1, c3+c4= 1}, and we have an infinite set of midpoints. If C∈equid(A, B), then c1+c2= (1 −c1) + (1 −c2), c3+c4= (1 −c3) + (1 −c4).Therefore c1+c2= 1, c3+c4= 1. Then mid(A, B) = equid(A, B). As we can see these sets coincide. Suppose that µC= (1 2,1 2), νC= (1 2,1 2). Then, d(A, B)=2, d(A, C)=2, d(C, B)=2. Thus, C∈equid(A, B) but C /∈segment(A, B), since d(A, C) + d(C, B)6=d(A, B), therefore equid(A, B)*segment(A, B) Figure 1. Midpoints between µA= (0,0), νA= (1,1) and µB= (1,1), νB= (0,0) Example 4.2. Now consider µA= (0,0), νA= (1,1), µB= (1,0), νB= (0,1), then d(A, B)=2, C∈mid(A, B)if and only if •c1+c2= 1 −c1+c2= 1 ⇒c1=1 2, c2=1 2, •1−c3+ 1 −c4=c3+ 1 −c4= 1 ⇒c3=1 2, c4=1 2, so mid(A, B) = n(1 2,1 2),(1 2,1 2)o. Also, C∈equid(A, B)if and only if •c1+c2= 1 −c1+c2⇒c1=1 2, •1−c3+ 1 −c4=c3+ 1 −c4⇒c3=1 2, then equid(A, B) = n(1 2, c2),(1 2, c4)o,hence mid(A, B)⊂equid(A, B). Figure 2. Equidistant points between µA= (0,0), νA= (1,1) and µB= (1,0), νB= (0,1) 38 DAVVAZ, HASSANI SADRABADI, NIETO AND TORRES 5. Intuitionistic fuzzy segments and intuitionistic fuzzy midpoints In this section, first we verify the segment between two points in a square and determine the set of midpoints between Aand B. Let µA= (a1, a2), νA= (a3, a4), µB= (b1, b2), νB= (b3, b4) be intuitionistic fuzzy subsets of the set X={x1, x2}. If µC= (c1, c2), νC= (c3, c4)∈segment(A, B), then: |a1−c1|+|a2−c2|+|b1−c1|+|b2−c2|=|a1−b1|+|a2−b2|,(5.1) |a3−c3|+|a4−c4|+|b3−c3|+|b4−c4|=|a3−b3|+|a4−b4|.(5.2) So, we obtain min{ai, bi} ≤ ci≤max{ai, bi}, i = 1,2,3,4. If min{ai, bi}> ci, then • |a1−c1|+|b1−c1|>|b1−a1|,|a2−c2|+|b2−c2|>|b2−a2|,so |a1−c1|+|a2−c2|+|b1−c1|+|b2−c2|>|b1−a1|+|b2−a2|that is contradiction with 5.1, also • |a3−c3|+|b3−c3|>|b3−a3|,|a4−c4|+|b4−c4|>|b4−a4|, therefore |a3−c3|+|a4−c4|+|b3−c3|+|b4−c4|>|b3−a3|+|b4−a4|,this relation is contradiction with 5.2. Theorem 5.1. The points in the segment between Aand Bis as follows: segment(A, B) = {(c1, c2),(c3, c4)|min{ai, bi} ≤ ci≤max{ai, bi}, i = 1,2,3,4}. Proof: Suppose that a1≤b1and a3≥b3(if a1> b1and a3≤b3we interchange the roles of µA, µB and νA, νB). Now, there are four possibilities, if µC= (c1, c2), νC= (c3, c4) then: (1) a1≤b1, a2≤b2, a3≥b3, a4≤b4⇒a1< c1< b1, a2< c2< b2, b3< c3< a3, a4< c4< b4, (2) a1≤b1, a2> b2, a3≥b3, a4≤b4⇒a1< c1< b1, b2< c2< a2, b3< c3< a3, a4< c4< b4, (3) a1≤b1, a2≤b2, a3≥b3, a4> b4⇒a1< c1< b1, a2< c2< b2, b3< c3< a3, b4< c4< a4, (4) a1≤b1, a2> b2, a3≥b3, a4> b4⇒a1< c1< b1, b2< c2< a2, b3< c3< a3, b4< c4< a4. Suppose that min{ai, bi} ≤ ci≤max{ai, bi}and we show C∈segment(A, B) that is: d(A, C) + d(C, B) = d(A, B). Proof: 1. If a1≤b1, a2≤b2, a3≥b3, a4≤b4, then d(A, C) = c1−a1+c2−a2+a3−c3+c4−a4, d(C, B) = b1−c1+b2−c2+c3−b3+b4−c4, d(A, C) = b1−a1+b2−a2+a3−b3+b4−a4. Therefore d(A, B) = d(A, C) + d(C, B). Similarly in all of cases 2,3,4, d(A, B) = d(A, C) + d(C, B). Theorem 5.2. The set of midpoints between Aand Bhas the following possibilities: (1) If either a1=b1,a3=b3or a2=b2,a4=b4, then there is a midpoint given by mid(A, B) = {µA+µB 2,νA+νB 2}. (2) If a1< b1, a2< b2, a3> b3and a4< b4, then the set of midpoints is as follows. If µC= (c1, c2),νC= (c3, c4), then: •c1+c2=1 2(a1+a2+b1+b2), •c3+c4=1 2(a3+a4+b3+b4), (3) If a1< b1, a2> b2, a3> b3, a4> b4, then: •c1−c2=1 2(a1−a2+b1−b2), •c3−c4=1 2(a3−a4+b3−b4), (4) If a1< b1, a2< b2, a3> b3, a4> b4, then: •c1+c2=1 2(a1+a2+b1+b2), TWIN HYPERCUBE FOR INTUITIONISTIC FUZZY SETS AND THEIR APPLICATION IN MEDICINE 39 Figure 3. Segment between Aand B: 1. a1≤b1, a2≤b2, a3≥b3, a4≤b4, 2. a1≤b1, a2> b2, a3≥b3, a4≤b4, 3. a1≤b1, a2≤b2, a3≥b3, a4> b4, 4. a1≤b1, a2> b2, a3≥b3, a4> b4. •c3−c4=1 2(a3−a4+b3−b4), (5) If a1< b1, a2> b2, a3> b3, a4< b4, then: •c1−c2=1 2(a1−a2+b1−b2), •c3+c4=1 2(a3+a4+b3+b4), Proof: If C∈mid(A, B), then d(A, C) = d(C, B) = 1 2d(A, B). So, we have: •d(A, C) = d(C, B), therefore: |a1−c1|+|a2−c2|+|a3−c3|+|a4−c4|=|b1−c1|+|b2−c2|+|b3−c3|+|b4−c4|,