Intuitionistic Fuzzy Sets: Spherical Representation and...

Intuitionistic Fuzzy Sets: SphericalRepresentation and DistancesY. Yang†, F. Chiclana∗

Centre for Computational Intelligence, School of Computing, De MontfortUniversity, Leicester LE1 9BH, UK

Most existing distances between intuitionistic fuzzy sets are defined in linear plane representationsin 2D or 3D space. Here, we define a new interpretation of intuitionistic fuzzy sets as a restrictedspherical surface in 3D space. A new spherical distance for intuitionistic fuzzy sets is introduced.We prove that the spherical distance is different from those existing distances in that it is nonlinearwith respect to the change of the corresponding fuzzy membership degrees. C© 2009 WileyPeriodicals, Inc.

1. INTRODUCTION

Research in cognition science1 has shown that people are faster at identifying anobject that is significantly different from other objects than at identifying an objectsimilar to others. The semantic distance between objects plays a significant role inthe performance of these comparisons.2 For the concepts represented by fuzzy setsand intuitionistic fuzzy sets, an element with full membership (nonmembership)is usually much easier to be determined because of its categorical difference fromother elements. This requires the distance between intuitionistic fuzzy sets or fuzzysets to reflect the semantic context of where the membership/nonmembership valuesare, rather than a simple relative difference between them.

Most existing distances based on the linear representation of intuitionistic fuzzysets are linear in nature, in the sense of being based on the relative difference betweenmembership degrees.3−15 Obviously, in some semantic contexts these distancesmight not seem to be the most appropriate ones. In such cases, nonlinear distancesbetween intuitionistic fuzzy sets may be more adequate to capture the semanticdifference. Here, new nonlinear distances between two intuitionistic fuzzy sets areintroduced. We call these distances spherical distances because their definition isbased on a spherical representation of intuitionistic fuzzy sets.

∗Author to whom all correspondence should be addressed: e-mail: [email protected].†e-mail: [email protected].

INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, VOL. 24, 399–420 (2009)C© 2009 Wiley Periodicals, Inc. Published online in Wiley InterScience(www.interscience.wiley.com). • DOI 10.1002/int.20342

400 YANG AND CHICLANA

The paper is set out as follows. In the following section, some preliminarydefinitions and notation on intuitionistic fuzzy sets needed throughout the paperare provided. In Section 3, a review of the existing geometrical interpretations ofintuitionistic fuzzy sets and the distance functions proposed and usually used inthe literature is given. The spherical interpretation of intuitionistic fuzzy sets andspherical distance functions are introduced in Section 4. Because fuzzy sets areparticular cases of intuitionistic fuzzy sets, the corresponding spherical distancefunctions for fuzzy sets are derived in Section 5. Finally, Section 6 presents ourconclusion.

2. INTUITIONISTIC FUZZY SETS: PRELIMINARIES

Intuitionistic fuzzy sets were introduced by Atanassov in Ref. 3. The followingprovides its definition, which will be needed throughout the paper:

DEFINITION 1. (Intuitionistic fuzzy set) An intuitionistic fuzzy set A of the universeof discourse U is given by

A = {〈u, μA(u), νA(u)〉 |u ∈ U}

where

μA : U → [0, 1] , νA : U → [0, 1]

and

0 ≤ μA(u) + νA(u) ≤ 1 ∀u ∈ U.

For each u, the numbers μA(u) and νA(u) are the degree of membership and degreeof nonmembership of u to A, respectively.

Another concept related to intuitionistic fuzzy sets is the hesitancy degree,τA(u) = 1 − μA(u) − νA(u), which represents the hesitance to the membership ofu to A.

We note that the main difference between intuitionistic fuzzy sets and traditionalfuzzy sets resides in the use of two parameters for membership degrees instead ofa single value. Obviously, μA(u) represents the lowest degree of u belonging to A,and νA(u) gives the lowest degree of u not belonging to A. If we consider μ = v− asmembership and ν = 1 − v+ as nonmembership, then we come up with the so-calledinterval-valued fuzzy sets [v−, v+].16

DEFINITION 2. (Interval-valued fuzzy set) An interval-valued fuzzy set A of theuniverse of discourse U is given by

A = {〈u, MA(u)〉 |u ∈ U}International Journal of Intelligent Systems DOI 10.1002/int

INTUITIONISTIC FUZZY SETS 401

where the function

MA : U → P[0, 1]

defines the degree of membership of an element u to A, being

P[0, 1] = {[a, b], a, b ∈ [0, 1], a ≤ b}

.Interval-valued fuzzy sets16,17 and intuitionistic fuzzy sets3−5 emerged from

different grounds and thus they have associated different semantics.18 However,they have been proven to be mathematically equivalent.19−23 and because of this wedo not distinguish them throughout this paper. For the sake of convenience whencomparing existing distances, we will apply the notation of intuitionistic fuzzy sets.Our conclusions could easily be adapted to interval-valued fuzzy sets.

3. EXISTING GEOMETRICAL REPRESENTATIONS OF ANDDISTANCES BETWEEN INTUITIONISTIC FUZZY SETS

In contrast to traditional fuzzy sets where only a single number is used torepresent membership degree, more parameters are needed for intuitionistic fuzzysets. Geometrical interpretations have been associated with these parameters,24

which are especially useful when studying the similarity or distance between sets.One of these geometrical interpretations of intuitionistic fuzzy sets was given byAtanassov in Ref. 5, as shown in Figure 1a, where a universe U and subset OST inthe Euclidean plane with Cartesian coordinates are represented. According to thisinterpretation, given an intuitionistic fuzzy set A, a function fA from U to OST canbe constructed such that if u ∈ U , then

P = fA(u) ∈ OST

is the point with coordinates 〈μA(u), νA(u)〉 for which 0 ≤ μA(u) ≤ 1, 0 ≤ νA(u) ≤1, 0 ≤ μA(u) + νA(u) ≤ 1.

We note that the triangle OST in Figure 1a is an orthogonal projection of the3D representation proposed by Szmidt and Kacprzyk,10 as shown in Figure 1b. InFigure 1b, in addition to μA(u) and νA(u), a third dimension is present, τA(u) =1 − μA(u) − νA(u). Because μA(u) + νA(u) + τA(u) = 1, the restricted plane RST

can be interpreted as the 3D counterpart of an intuitionistic fuzzy set. Therefore, ina similar way to Atanassov’s procedure, for an intuitionistic fuzzy set A a functionfA from U to RST can be constructed in such a way that given u ∈ U , then

S = fA(u) ∈ RST

has coordinates 〈μA(u), νA(u), τA(u)〉 for which 0 ≤ μA(u) ≤ 1, 0 ≤ νA(u) ≤ 1,0 ≤ τA(u) ≤ 1 and μA(u) + νA(u) + τA(u) = 1.

International Journal of Intelligent Systems DOI 10.1002/int


Figure 1. 2D and 3D representations of intuitionistic fuzzy sets.

A fuzzy set is a special case of an intuitionistic fuzzy set where τA(x) = 0holds for all elements. In this case, both OST and RST in Figure 1 converge to thesegment ST . Therefore, under this interpretation, the distance between two fuzzysets is based on the membership functions, which depends on just one parameter(membership). Given any two fuzzy subsets A = {〈ui, μA(ui)〉 : ui ∈ U} and B ={〈ui, μB(ui)〉 : ui ∈ U} with U = {u1, u2, . . . , un}, the following distances havebeen proposed8:

• Hamming distance d1(A, B)

d1(A, B) =n∑

i=1

|μA(ui) − μB (ui)|

• Normalized Hamming distance l1(A, B)

l1(A, B) = 1

n

n∑i=1

|μA(ui) − μB (ui)|

• Euclidean distance e1(A, B)

e1(A, B) =√√√√ n∑

i=1

(μA(ui) − μB (ui))2

• Normalized Euclidean distance q1(A, B)

q1(A, B) =√√√√ 1

n

n∑i=1

(μA(ui) − μB (ui))2



In the case of intuitionistic fuzzy sets, different distances have been defined ac-cording to either of the 2D or 3D interpretations. Atanassov in Ref. 5 presents the fol-lowing distances for any two intuitionistic fuzzy subsets A = {〈ui, μA(ui), νA(ui)〉 :ui ∈ U} and B = {〈ui, μB(ui), νB(ui)〉 : ui ∈ U} using the above 2D interpretation:


d2(A,B) = 1

2

n∑i=1

[|μA(ui) − μB (ui)| + |νA(ui) − νB (ui)|]

• Normalized Hamming distance l2(A,B)

l2(A,B) = 1

2n

n∑i=1

[|μA(ui) − μB (ui)| + |νA(ui) − νB (ui)|]

• Euclidean distance e2(A,B)

e2(A,B) =√√√√1

2

n∑i=1

[(μA(ui) − μB (ui))2 + (νA(ui) − νB (ui))2]


q2(A, B) =√√√√ 1

2n

n∑i=1

[(μA(ui) − μB (ui))2 + (νA(ui) − νB (ui))2]

Based on the 3D representation, Szmidt and Kacprzyk9,10 modified these dis-tances to include the third parameter τA(u):


d3(A, B) = 1

2

n∑i=1

[|μA(ui) − μB (ui)| + |νA(ui) − νB (ui)| + |τA(ui) − τB (ui)|]

• Normalized Hamming distance l3(A,B)

l3(A, B) = 1

2n

n∑i=1

[|μA(ui) − μB (ui)| + |νA(ui) − νB (ui)| + |τA(ui) − τB (ui)|]

• Euclidean distance e3(A,B)

e3(A, B) =√√√√1

2

n∑i=1

[(μA(ui) − μB (ui))2 + (νA(ui) − νB (ui))2 + (τA(ui) − τB (ui))2]




q3(A, B) =√√√√ 1

2n

n∑i=1

[(μA(ui) − μB (ui))2 + (νA(ui) − νB (ui))2 + (τA(ui) − τB (ui))2]

All the above distances clearly adopt the linear plane interpretation, as shownin Figure 1, and therefore they reflect only the relative differences between thememberships, nonmemberships, and hesitancy degrees of intuitionistic fuzzy sets.The following lemma illustrates this characteristic:

LEMMA 1. Let A = {〈ui, μA(ui), νA(ui)〉 : ui ∈ U}, B = {〈ui, μB(ui), νB(ui)〉 :ui ∈ U}, C = {〈ui, μC(ui), νC(ui)〉 : ui ∈ U}, and G = {〈ui, μG(ui), νG(ui)〉 : ui

∈U} be four intuitionistic fuzzy sets of the universe of discourse U = {u1, u2, . . . , un}.If the following conditions hold

μA(ui) − μB(ui) = μC(ui) − μG(ui)

νA(ui) − νB(ui) = νC(ui) − νG(ui)

then

D(A, B) = D(C, G)

being D any of the above Atanassov’s 2D or Szmidt and Kacprzyk’s 3D distancefunctions.

Proof. The proof is obvious for Atanassov’s 2D distances. For Szmidt andKacprzyk’s 3D distances, the proof follows from the fact that

μA(ui) − μB(ui) = μC(ui) − μG(ui) ∧ νA(ui) − νB(ui) = νC(ui) − νG(ui)

imply

τA(ui) − τB(ui) = τC(ui) − τG(ui)

�

If |τA(ui) − τB(ui)| = |τC(ui) − τG(ui)|, then conditions in Lemma 1 can begeneralized to

|μA(ui) − μB(ui)| = |μC(ui) − μG(ui)|

|νA(ui) − νB(ui)| = |νC(ui) − νG(ui)|International Journal of Intelligent Systems DOI 10.1002/int


This means that if we move both sets in the space shown in Figure 1b with the samechanges in membership, nonmembership, and hesitancy degrees, then we obtainexactly the same distance between the two fuzzy sets. This linear feature of theabove distances may not be adequate in some cases, because the human perceptionis not necessarily always linear. For example, we can classify the human behavioras perfect, good, acceptable, poor, and worst. Using fuzzy sets, we can assign theirfuzzy membership as 1, 0.75, 0.5, 0.25, and 0. To find out if someone’s behavior isperfect or not, we only need to check if there is anything wrong with him. However,to differentiate good from acceptable, we have to count their positive and negativepoints. Obviously, the semantic distance between perfect and good should be greaterthan the semantic distance between good and acceptable. This semantic difference isnot shown by the linear distances between their memberships. Therefore, a nonlinearrepresentation of the distance between two intuitionistic fuzzy sets may benefit therepresentative power of intuitionistic fuzzy sets. Although nonlinearity could bemodeled by using many different expressions, we will consider and use a simpleone to model it. Here, we propose a new geometrical interpretation of intuitionisticfuzzy sets in 3D space using a restricted spherical surface. This new representationprovides a convenient nonlinear measure of the distance between two intuitionisticfuzzy sets.

4. SPHERICAL INTERPRETATION OF INTUITIONISTIC FUZZYSETS: SPHERICAL DISTANCE

Let A = {〈u, μA(u), νA(u)〉 : u ∈ U} be an intuitionistic fuzzy set. We have

μA(u) + νA(u) + τA(u) = 1

which can be equivalently transformed to

x2 + y2 + z2 = 1

with

x2 = μA(u), y2 = νA(u), z2 = τA(x)

It is obvious that this is not the only transformation that could be used. However,as shown in the existing distances, there is no special reason to discriminate μA(u),νA(u), and τA(u). Therefore, a simple nonlinear transformation to unit sphere in a3D Euclidean space is selected here, as shown in Figure 2.

This transformation allows us to interpret an intuitionistic fuzzy set as a re-stricted spherical surface. An immediate consequence of this interpretation is thatthe distance between two elements of an intuitionistic fuzzy set can be defined asthe spherical distance between their corresponding points on its restricted sphericalsurface representation. This distance is defined as the shortest path between the two



Figure 2. 3D sphere representation of intuitionistic fuzzy sets.

points, i.e. the length of the arc of the great circle passing through both points. Forpoints P and Q in Figure 2, their spherical distance is25.

ds(P, Q) = arccos

{1 − 1

2

[(xP − xQ)2 + (yP − yQ)2 + (zP − zQ)2

]}.

This expression can be used to obtain the spherical distance between two in-tuitionistic fuzzy sets, A = {〈ui, μA(ui), νA(ui)〉 : ui ∈ U} and B = {〈ui, μB(ui),νB(ui)〉 : ui ∈ U} of the universe of discourse U = {u1, u2, . . . , un}, as follows:

ds(A, B) = 2

π

n∑i=1

arccos

{1 − 1

2

[(√μA(ui) −

√μB(ui)

)2

+(√νA(ui) −

√νB(ui)

)2 + (√τA(ui) −

√τB(ui)

)2]}

where the factor 2π

is introduced to get distance values in the range [0, 1] instead of[0, π

2 ]. Because μA(ui) + νA(ui) + τA(ui) = 1 and μB(ui) + νB(ui) + τB(ui) = 1,we have that

ds(A, B) = 2

π

n∑i=1

arccos(√

μA(ui)μB(ui) +√

νA(ui)νB(ui) +√

τA(ui)τB(ui))

This is summarized in the following definition:



DEFINITION 3. (Spherical distance) Let A = {〈ui, μA(ui), νA(ui)〉 : ui ∈ U} andB = {〈ui, μB(ui), νB(ui)〉 : ui ∈ U} be two intuitionistic fuzzy sets of the universe ofdiscourse U = {u1, u2, . . . , un}. The spherical and normalized spherical distancesbetween A and B are

• Spherical distance ds(A, B)

ds(A, B) = 2

π

n∑i=1

arccos(√

μA(ui)μB (ui) +√

νA(ui)νB (ui) +√

τA(ui)τB (ui))

• Normalized spherical distance dns(A, B)

dns(A,B) = 2

nπ

n∑i=1

arccos(√

μA(ui)μB (ui) +√

νA(ui)νB (ui) +√

τA(ui)τB (ui))

Clearly, we have that 0 ≤ ds(A, B) ≤ n and 0 ≤ dns(A, B) ≤ 1.

Different from the distances in Section 3, the proposed spherical distancesimplement in their definition not only the difference between membership, non-membership, and hesitancy degrees but also their actual values. This is shown in thefollowing result:

LEMMA 2. Let A = {〈ui, μA(ui), νA(ui)〉 : ui ∈ U} and B = {〈ui, μB(ui), νB(ui)〉 :ui ∈ U} be two intuitionistic fuzzy subsets of the universe of discourse U = {u1,

u2, . . . , un}. Let a = {a1, a2, . . . , an} and b = {b1, b2, . . . , bn} be two sets of realnumbers (constants). If the following conditions hold for each ui ∈ U :

μB(ui) = μA(ui) + ai

νB(ui) = νA(ui) + bi

then the following inequalities hold:

2

π

n∑i=1

arccos√

1 − e2i ≤ ds(A, B) ≤ 2

π

n∑i=1

arccos√

(1 − ci)(1 − |ai | − |bi |)

2

nπ

n∑i=1

arccos√

1 − e2i ≤ dns(A, B) ≤ 2

nπ

n∑i=1

arccos√

(1 − ci)(1 − |ai | − |bi |)



where ci = max{|ai |, |bi |} and ei = min{|ai |, |bi |}. The maximum distance betweenA and B is obtained if and only if one of them is a fuzzy set or their availableinformation supports only opposite membership degree for each one.

Proof. See Appendix A. �

Owing to the nonlinear characteristic of the spherical distance, they do notsatisfy Lemma 1. However, the following properties hold for the spherical distances:

LEMMA 3. Let A = {〈ui, μA(ui), νA(ui)〉 :ui ∈ U}, B = {〈ui, μB(ui), νB(ui)〉 : ui ∈U} and E = {〈ui, μE(ui), νE(ui)〉 : ui ∈ U} be three intuitionistic fuzzy subsetsof the universe of discourse U = {u1, u2, . . . , un}. Let a = {a1, a2, . . . , an} andb = {b1, b2, . . . , bn} two sets of real positive numbers (constants) satisfying thefollowing conditions:

|μB(ui) − μA(ui)| = ai, |νB(ui) − νA(ui)| = bi

|μE(ui) − μA(ui)| = ai, |νE(ui) − νA(ui)| = bi

If E is one of the two extreme crisp sets with either μE(ui) = 1 or νE(ui) = 1 forall ui ∈ U , then the following inequalities hold:

ds(A, B) < ds(A, E), dns(A, B) < dns(A, E)

The distance between intuitionistic fuzzy sets A and B is always lower than thedistance between A and the extreme crisp sets E under the same difference of theirmemberships and nonmemberships.

Proof. We provide the proof just for the extreme fuzzy set with full memberships,being the proof for full nonmembership similar.

With E being the extreme crisp set with full memberships, we have

μE(ui) = 1, νE(ui) = 0, τE(ui) = 0

Because |μE(ui) − μA(ui)| = ai and |νE(ui) − νA(ui)| = bi , then

μA(ui) = 1 − ai, νA(ui) = bi, τA(ui) = ai − bi

From |μB(ui) − μA(ui)| = ai and |νB(ui) − νA(ui)| = bi , it is

μB(ui) = 1 − 2ai, νB(ui) = 2bi, τB(ui) = 2(ai − bi).



Therefore,

ds(A, B) = 2

π

n∑i=1

arccos

(√(1 − ai)(1 − 2ai) +

√2b2

i +√

2(ai − bi)2

)

= 2

π

n∑i=1

arccos(√

2ai +√

(1 − ai)(1 − 2ai))

and

ds(A, E) = 2

π

n∑i=1

arccos(√

1 − ai

)

Obviously, we have

√2ai +

√(1 − ai)(1 − 2ai) >

√1 − ai

Thus

ds(A, B) < ds(A, E)

and dividing by n

dns(A, B) < dns(A, E)�

Lemma 3 shows that the extreme crisp sets with full memberships or full non-memberships are categorically different from other intuitionitic fuzzy sets. Withthe same difference of memberships and nonmemberships, the distance from an ex-treme crip set is always greater than the distances from other intuitionistic fuzzy sets.This conclusion agrees with our human perception about the quality change againstquantity change and captures the semantic difference between extreme situation andintermediate situations.

5. SPHERICAL DISTANCES FOR FUZZY SETS

As we have already mentioned, fuzzy sets are particular cases of intuitionisticfuzzy sets. Therefore, the above spherical distances can be applied to fuzzy sets. Inthe following we provide Lemma 4 for the distance between two fuzzy sets.

LEMMA 4. Let A = {〈ui, μA(ui)〉 : ui ∈ U} and B = {〈ui, μB(ui)〉 : ui ∈ U} be twofuzzy sets in the universe of discourse U = {u1, u2, . . . , un}. Let a = {a1, a2, . . . , an}be a set of nonnegative real constants. If |μA(ui) − μB(ui)| = ai holds for each



ui ∈ U , then the following inequalities hold:

2

π

n∑i=1

arccos√

1 − ai2 ≤ ds(A, B) ≤ 2

π

n∑i=1

arccos√

1 − ai

2

nπ

n∑i=1

arccos√

1 − ai2 ≤ dns(A, B) ≤ 2

nπ

n∑i=1

arccos√

1 − ai

The maximum distance between A and B is achieved if and only if one of them is acrisp set.

Proof. According to Definition 3, we have

ds(A, B) = 2

π

n∑i=1

arccos(√

μA(ui)μB(ui) +√

νA(ui)νB(ui) +√

τA(ui)τB(ui))

For fuzzy sets, we have

τA(ui) = 0 and νA(ui) = 1 − μA(ui)

τB(ui) = 0 and νB(ui) = 1 − μB(ui)

Because |μA(ui) − μB(ui)| = ai and μA(ui) ≥ 0, we have

μB(ui) = μA(ui) ± ai

If


then

ds(A, B) = 2

π

n∑i=1

arccos(√

μA(ui)μB(ui) +√

(1 − μA(ui))(1 − μB(ui))

= 2

π

n∑i=1

arccos(√

μA(ui)(μA(ui) + ai)

+√

(1 − μA(ui))(1 − μA(ui) − ai))



This can be rewritten as

ds(A, B) = 2

π

n∑i=1

arccos fi(μA(ui))

with fi(t) = √t(t + ai) + √

(1 − t)(1 − t − ai), t ∈ [0, 1 − ai]. The extremes offunction fi(t) will be among the solution of

f ′i (t) = 0 t ∈ (0, 1 − ai)

and the values 0 and 1 − ai , i.e, among (1 − ai)/2, 0 and 1 − ai . The maximum value√1 − ai

2 is obtained when t = (1 − ai)/2, whereas the minimum value√

1 − ai isobtained in both 0 and 1 − ai . We conclude that

2

π

n∑i=1

arccos√

1 − ai2 ≤ ds(A, B) ≤ 2

π

n∑i=1

arccos√

1 − ai

When t = 0 or t = 1 − ai , we have, respectively, μA(ui) = 0 and μB(ui) = 1 −ai + ai = 1, which implies that one set among A and B has to be crisp in order toreach the maximum value under the given difference in their membership degrees.

Following a similar reasoning, it is easy to prove that the same conclusion isobtained in the case μB(ui) = μA(ui) − ai . If μB(ui) = μA(ui) − ai for some i,and μB(uj ) = μA(uj ) + aj for some j , then we could separate the elements intotwo different groups, each of them satisfying the inequalities, and therefore theirsummation obviously satisfying it too. The normalized inequality is obtained justby dividing the first one by n. �

Because spherical distances are quite different from the traditional distances,the semantics associated with them also differ. For the same relative difference inmembership degrees, the spherical distance varies with the locations of its two rele-vant sets in the membership degree space, 2D for fuzzy sets and 3D for intuitionisticfuzzy sets. The spherical distance achieves its maximum when one of the fuzzy setsis an extreme crisp set. The following example illustrates this effect:

Example 1. Consider our previous example on human behavior. We can classifyour behavior as perfect, good, acceptable, poor, and worst, with corresponding fuzzymembership of 1, 0.75, 0.5, 0.25, and 0, respectively. Let A = {〈u, 0.75〉 : u ∈ U},B = {〈u, 0.5〉 : u ∈ U}, and E = {〈u, 1〉 : u ∈ U} be three fuzzy subsets, and U ={u} is a universe of discourse with one element only.

From Section 3, we have

d1(A, B) = l1(A, B) = e1(A, B) = q1(A, B) = 0.25

d1(A, E) = l1(A, E) = e1(A, E) = q1(A, E) = 0.25



Obviously, we have

d1(A, B) = d1(A, E), l1(A, B) = l1(A, E)

e1(A, B) = e1(A, E), q1(A, B) = q1(A, E)

From Definition 3, we have

dns(A, B) = ds(A, B)

= 2

π

n∑i=1

arccos(√

0.75 ∗ 0.5 +√

(1 − 0.75)(1 − 0.5))

= 0.17

dns(A, E) = ds(A, E) = 2

π

n∑i=1

arccos(√

0.75 ∗ 1)

= 0.55

We note that the traditional linear distance of fuzzy sets does not differentiate thesemantic difference of a crisp set from a fuzzy set. However, ds(A, E) and dns(A, E)are much greater than ds(A, B) and dns(A, B). This demonstrates that the crisp setE is much more different from A than B although their membership difference arethe same. Hence, the proposed spherical distance does show the semantic differencebetween a crisp set and a fuzzy set. This is useful when this kind of semanticdifference is significant in the consideration.

Figure 3 shows four comparisons between the spherical distance and the Ham-ming distance for two fuzzy subsets A, B with a universe of discourse with oneelement U = {u}. The curves represent the spherical distance, and straight linesdenote Hamming distances. Figure 3a displays how the distance changes with re-spect to μB(x) when μA(x) = 0; Figure 3b uses the value μA(x) = 1; in Figure 3cthe value μA(x) = 0.2 is used; finally μA(x) = 0.5 is used in Figure 3d. Clearly,the spherical distance changes sharply for values close to the two lower and uppermemberships values, but slightly for values close to the middle membership value.In the case of the Hamming distance, the same rate of change is always obtained.

Figure 4 displays how the spherical distance and Hamming distance changeswith respect to μB(x) for all possible values of μA(x). Figure 4a shows that thespherical distance forms a curve surface, whereas a plane surface produced by theHamming distance is shown in Figure 4b. Their contours in the bottom clearly showtheir differences. The contours for spherical distances are ellipses coming from (0, 0)and (1, 1) with curvatures increasing sharply near (0, 1) and (1, 0). Compared withthese ellipses, the contours of Hamming distance are a set of parallel lines. Thesefigures prove our conclusions in Lemma 4: The spherical distances do not remainconstant as Hamming distances do when both sets experiment a same change intheir membership degrees.



Figure 3. Comparison between spherical and Hamming distances for fuzzy sets.

Figure 4. Grid of spherical and Hamming distances for fuzzy sets.

6. CONCLUSIONS

An important issue related with the representation of intuitionistic fuzzy setsis that of measuring distances. Most existing distances are based on linear plane



representation of intuitionistic fuzzy sets, and therefore are also linear in nature, inthe sense of being based on the relative difference between membership degrees.In this paper, we have looked at the issue of 3D representation of intuitionisticfuzzy sets. We have introduced a new spherical representation, which allowed usto define a new distance function between intuitionistic fuzzy sets: the sphericaldistance. We have shown that the spherical distance is different from those existingdistances in that it is nonlinear with respect to the change of the corresponding fuzzymembership degrees, and thus it seems more appropriate than usual linear distancesfor non linear contexts in 3D spaces.

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APPENDIX: PROOF FOR LEMMA 2

According to Definition 3, we have

ds(A, B) = 2

π

n∑i=1

arccos(√

μA(ui)μB(ui) +√

νA(ui)νB(ui) +√

τA(ui)τB(ui))

Because A and B satisfy


νB(ui) = νA(ui) + bi

then

τB(ui) = 1 − μA(ui) − νA(ui) − ai − bi

and

ds(A, B) = 2

π

n∑i=1

arccos(√

μA(ui)(μA(ui) + ai) +√

νA(ui)(νA(ui) + bi)

+√(1 − μA(ui) − νA(ui))(1 − μA(ui) − νA(ui) − ai − bi)

)

Let

f (u, v) =√

u(u + ai) +√

v(v + bi) +√

(1 − u − v)(1 − u − v − ai − bi).

Denoting a = |ai | and b = |bi |, then we distinguish 4 possible cases:Case 1: ai ≥ 0 and bi ≥ 0. In this case,

f (u, v) =√

u(u + a) +√

v(v + b) +√

(1 − u − v)(1 − u − v − a − b).



Let f ′(u, v)|u = 0, we have

u = a(v − 1)

band v = a(1 − a − b − v)

2a + b.

Let f ′(u, v)|v = 0, then

v = b(u − 1)

aand v = b(1 − a − b − u)

2b + a.

Because

u = a(v − 1)

b< 0 and v = b(u − 1)

a< 0,

the valid solutions are

u = a(1 − a − b − v)

2a + band v = b(1 − a − b − u)

2b + a.

Solving these equations, we have

u = a(1 − a − b)

2(a + b)and v = b(1 − a − b)

2(a + b),

and therefore

f0 = f

(a(1 − a − b)

2(a + b),b(1 − a − b)

2(a + b)

)=

√1 − (a + b)2.

Obviously, the third square root in f (u, v) must be defined so

0 ≤ u ≤ 1 − a − b and 0 ≤ v ≤ 1 − a − b.

The boundary points are reached when u and v get their minimum or maximumvalues. Denoting t = τA(ui), the following needs to be verified

u + v + t = 1.

If u = v = 1 − a − b then t = 2a + 2b − 1, and (1 − u − v)(1 − u − v − a − b) =(2a + 2b − 1)(a + b − 1) ≤ 0, which implies that the third square root in f (u, v)is not defined. Therefore, we have three boundary points for A

u = 0, v = 1 − a − b, t = a + b

u = 1 − a − b, v = 0, t = a + b



u = 0, v = 0, t = 1,

and therefore

f1 = f (0, 1 − a − b) =√

(1 − a)(1 − a − b)

f2 = f (1 − a − b, 0) =√

(1 − b)(1 − a − b)

f3 = f (0, 0) = √1 − a − b.

If ai ≥ 0 and bi ≥ 0, then a + b ≤ 1, and we have

f1 ≤ f3 ≤ f0

f2 ≤ f3 ≤ f0.

The relationship between f1 and f2 depends on the relationship between a and b.Let c = max{a, b} and e = min{a, b}, then we have

√(1 − c)(1 − a − b) ≤ f (u, v) ≤

√1 − e2.

Case 2: ai ≤ 0 and bi ≤ 0. In this case, following a similar reasoning to theabove one, the same conclusion is obtained.

Case 3: ai ≤ 0 and bi ≥ 0. In this case

f (u, v) =√

u(u − a) +√

v(v + b) +√

(1 − u − v)(1 − u − v + a − b).

Let f ′(u, v)|u = 0, then

u = a(1 − v)

band v = a(1 + a − b − v)

2a − b.

Let f ′(u, v)|v = 0, then

v = b(1 − u)

aand v = −b(1 + a − b − u)

a − 2b.

For u = a(1 − v)/b and v = b(1 − u)/a, we get a = b, hence u + v = 1, and thenboth A and B are fuzzy sets. According to Lemma 4, we know that

√1 − a ≤ f (u, v) ≤

√1 − a2.

Note that the following also holds: (1 − a)(1 − a − b) ≤ (1 − a).



For u = [a(1 + a − b − v)]/(2a − b) and v = [−b(1 + a − b − u)]/(a − 2b),we have

u = a(1 + a − b)

2(a − b)and v = −b(1 + a − b)

2(a − b).

If a > b then u ≥ 0 and v ≤ 0, and therefore the only possibility here is v = 0, i.eb = 0 in which case it is

f0 = f

(1 + a

2, 0

)=

√1 − a2.

If a < b then u ≤ 0 and v ≥ 0, and therefore a = 0, in which case it is

f1 = f

(0,

1 − b

2

)=

√1 − b2.

For u = a(1 − v)/b and v = [−b(1 + a − b − u)]/(a − 2b), we have

u = a(1 + b)

2band v = 1 − b

2

and we obtain

f

(a(1 + b)

2b,

1 − b

2

)=

√1 − b2 = f1.

Also, because u + v ≤ 1 we have that (a − b) · (1 + b) ≤ 0 and, therefore, it isa ≤ b.

For u = [a(1 + a − b − v)]/(2a − b) and v = b(1 − u)/a, we have

u = 1 + a

2and v = b(1 − a)

2a

and

f

(1 + a

2,b(1 − a)

2a

)=

√1 − a2 = f0.

In this case it is a ≥ b.

The ranges for u and v are

a ≤ u ≤ 1 and 0 ≤ v ≤ 1 − max{a, b}

therefore, we have the following boundary points for A

u = a, v = 1 − a, t = 0



u = a, v = 1 − b, t = b − a

u = 1, v = 0, t = 0

u = a, v = 0, t = 1 − a

Thus

f2 = f (a, 1 − a) =√

(1 − a)(1 − a + b)

f3 = f (a, 1 − b) = √1 − b

f4 = f (1, 0) = √1 − a

f5 = f (a, 0) =√

(1 − a)(1 − b)

The following inequalities hold

f2 ≤ f4 ≤ f0

f5 ≤ f4 ≤ f0

f5 ≤ f3 ≤ f1

Denoting

fmin =√

(1 − c)(1 − a − b) and fmax =√

1 − e2

with e = min{a, b} and c = max{a, b}, then we have

√1 − e2 ≥ f (u, v) ≥

√(1 − c)(1 − a − b).

Case 4: ai ≥ 0 and bi ≤ 0. In this case, following a similar reasoning to theabove one, the same conclusion is obtained.

In the four cases, we conclude that

2

π

n∑i=1

arccos√

1 − e2i ≤ ds(A, B) ≤ 2

π

n∑i=1

arccos√

(1 − ci)(1 − |ai | − |bi |)

2

nπ

n∑i=1

arccos√

1 − e2i ≤ ds(A, B) ≤ 2

nπ

n∑i=1

arccos√

(1 − ci)(1 − |ai | − |bi |)

where ci = max{|ai |, |bi |} and ei = min{|ai |, |bi |}.International Journal of Intelligent Systems DOI 10.1002/int


According to the boundary points discussed in the above four cases, if aibi ≥ 0holds for each ui , we have

μA(ui) = 0, νA(ui) = 1 − a − b, τA(ui) = a + b

or

μA(ui) = 1 − a − b, νA(ui) = 0, τA(ui) = a + b

hence, set B has to satisfy

μB(ui) = a, νB(ui) = 1 − a, τB(ui) = 0

or

μB(ui) = 1 − b, νB(ui) = b, τB(ui) = 0

Obviously, B is a fuzzy set. A similar conclusion can be drawn if aibi < 0 holdsfor each ui and

√(1 − c)(1 − c + e) ≤ √

(1 − a)(1 − b). However, if ai < 0 andbi > 0 hold for each ui and

√(1 − a)(1 − a + b) >

√(1 − a)(1 − b), we have

μA(ui) = a, νA(ui) = 0, τA(ui) = 1 − a

hence, the set B satisfies

μB(ui) = 0, νB(ui) = b, τB(ui) = 1 − b

Obviously, A is an intuitionistic fuzzy set with available information supportingonly membership, whereas B is an intuitionistic fuzzy set with available informationsupporting only nonmembership.


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