Measuring Contradiction between two AIFS

MEASURING CONTRADICTION BETWEEN TWO AIFS∗

C. TORRES-BLANC†_{, E. E. CASTI ˜NEIRA, S. CUBILLO}

Dpto. Matem´atica Aplicada Universidad Polit´ecnica de Madrid 28660 Boadilla del Monte. Madrid (Spain)

{ctorres, ecastineira, scubillo}@fi.upm.es

This paper is devoted to introduce an axiomatic model to distinguish what functions are suitable for measuring the degree of contradiction between two Atanassov’s intuitionistic fuzzy sets. After stating the needed background, in section 2, we justify and present the axioms that a contradiction measure must satisfy, and the first examples are set out. After motivating the necessity of achieving some definition for modelling the continuity, in the next section we introduce the concepts of semicontinuity from below and semicontinuity from above for contradiction measures. Finally, in section 4, some families of con-tradiction measures are constructed.

Keywords: Atanassov’s Intuitionistic fuzzy sets; contradiction measures; con-tinuity from below and from above.

1. Preliminaries

As it is well known, an Atanassov’s intuitionistic fuzzy set (AIFS) on a universe X is a set A={(x, µA(x), νA(x))|x ∈ X}, where µA, νA:X →

[0,1] satisfy that µA(x) +νA(x) ≤ 1 for all x ∈ X (see1). Hence, A is

an L-fuzzy set (see2_{), where L ={α= (α1, α2) ∈ [0,1]}2_{| α1+α2 ≤ 1}} is a bounded and complete lattice with the order defined by: (α1, α2)≤L (β1, β2) if and only if α1 ≤ β1 and α2 ≥ β2; and with lowest element, 0L = (0,1), and greatest element, 1L = (1,0). Thus χA = (µA, νA)∈LX

is theL-membership function ofA. Moreover, we say thatAisL-normalif Infx∈X(νA(x)) = 0 holds, and we denote LX0 ={χ∈LX |χisL-normal}.

Similarly to the fuzzy case, ifN is astrong intuitionistic fuzzy negation (S-IFN)(that is, N : L → L is a decreasing function with N(0L) = 1L,

∗_{This paper is partially supported by CICYT (Spain) under Project TIN}

2005-08943-C02-01.

N(1L) = 0L and N(N(α)) =αfor allα∈L; see3 or4), two AIFS Aand B, or alternativelyχA_andχB_{, areN-contradictory ifχ}A_(x)≤

L(N ◦χB)(x) for allx∈X. Also, AandB, orχA_{and χ}B_{, are contradictory if they are} N-contradictory regarding some S-IFNN (see5_).

If N is an S-IFN and N the strong fuzzy negation associated with N

according to the representation theorem of S-IFN (that is, N(α1, α2) = (N(1 − α2),1 − N(α1)), for all (α1, α2) ∈ L, see3 _and4_{), then χ}A

and χB _{are N-contradictory if and only if N(µ}

A(x)) +νB(x) ≥ 1 and

N(µB(x)) +νA(x) ≥ 1, for all x ∈ X (see5); furthermore, χA and χB

are N-contradictory if and only if g(µA(x)) +g(1 −νB(x)) ≤ 1 and

g(µB(x)) +g(1−νA(x)) ≤ 1, ∀x ∈ X, where g is the generator of N

(that is,N(α) =g−1_{(1−g(α)) for allα∈[0,1], see}6_).

In7_{, Cubillo et al. analyzed the regions of [0,1]}2 _{in which N} -contradictory sets are located, with the purpose of suggesting some way for measuring how contradictory two AIFS are. Given χA _{= (µ}

A, νA), χB =

(µB, νB) ∈ LX, if χAB = (µA, νB) and χBA = (µB, νA), then χA

and χB _{are N-contradictory if and only if χ}AB_{(X), χ}BA_{(X) ⊂ R}

N = ©

(α1, α2)∈[0,1]2_{|N(α1) +α2≥1}ª_{; and R}

N is called the region of N

-contradiction (see Figure 1).

(0,0)

N_-Contradiction Region

(1,1)

(a ,1-a )N N 0LI

1LI

c

c

Fig. 1. Region ofN-contradiction and theN-contradictory setsAandB. 2. Axioms for measuring contradiction between two AIFS

two AIFS is symmetrical since N is an S-IFN; so a contradiction mea-sure should be symmetrical. Finally, if A and B are two AIFS such that χA _≤

L χB (that is,χA(x)≤L χB(x) for all x∈X), then for any AIFSC that isN-contradictory withB, it isχA_≤

LχB ≤LN ◦χC, and thus “B is closer” to be non-contradictory withCthanA, which means thatBis less contradictory withC thanA; therefore the anti-monotonicity is a suitable requirement. These remarks lead us to the following definition.

Definition 2.1. LetX 6=∅, a functionC :LX_×LX _{→[0,1] is a measure}

of contradictionon AIFS, or onLX_{, if it satisfies:}

(c.i) C(χ0L_{, χ}0L_{) = 1, whereχ}0L _{is the empty AIFS:χ}0L_{(x) = 0}

L ∀x∈X. (c.ii) Given χA_{, χ}B_∈LX_{, ifχ}A _orχB _{isL-normal, thenC(χ}A_{, χ}B_{) = 0.}

(c.iii) Symmetry:C(χA_{, χ}B_{) =C(χ}B_{, χ}A_{) for allχ}A_{, χ}B _∈LX_.

(c.iv) Anti-monotonicity: GivenχA_{, χ}B _∈LX _{such that χ}A _≤

L χB, then C(χA_{, χ}C_{)≥ C(χ}B_{, χ}C_{), for allχ}C_∈LX_.

The set of all contradiction measures on LX _{will be denoted by} CM(LX_×LX_).

In7_{, it was proved that the previous axioms are satisfied by the following} functions: Let N be an S-IFN associated with N, and g the generator of N, ifχA_{= (µ}

A, νA), χB = (µB, νB)∈LX

CN(χA, χB) = Max (0,Min (FN(µA, νB), FN(µB, νA)))

Cg(χA, χB) = Max(0,Min(Fg(µA, νB), Fg(µB, νA)))

CE(χA, χB) =Min(d(χ

AB_(X),R∗

N),d(χBA(X),R∗N))

d(0L,R∗N)

(1)

where, for allµ, ν∈[0,1]X_,F

N(µ, ν) = Inf

x∈X(N(µ(x))+ν(x)−1),Fg(µ, ν) =

1−Sup

x∈X(g(µ(x)) +g(1−ν(x))),R

∗

N ={(α1, α2)∈[0,1]2|N(α1) +α2<1} anddthe euclidean distance (see Figure 2).

(0,0)

(1,1)

0

1

c

(

X

₎

c

(

X

₎

C

( , )

N

c

c

C

( , )

E

c

c

N_{(a )+a =1}

The previous definition does not guarantee any kind of continuity of the measures, as the following example shows: The functionCN :LX →[0,1], given byC(χA_{, χ}B_{) = 1 ifχ}A_=χB_=χ0L _andC(χA_{, χ}B_{) = 0 otherwise, is a}

measure of contradiction, that changes sharply in (χ0L, χ0L). So, if we want

to modelize the continuity, we need to impose some additional conditions.

3. Axioms for modelling the contradiction measure continuity

In order to demand a measure to change smoothly, we propose new axioms. Recall that S ⊂LX _{is a semilattice from below if for all χ}A_{, χ}B _{∈ S,}

Sup{χA_{, χ}B_{} ∈S is satisfied; and similarly, S ⊂L}X _{is a semilattice from}

above if for allχA_{, χ}B_{∈S, Inf{χ}A_{, χ}B_{} ∈S is satisfied (see} 8 _or 9_).

Definition 3.1. LetX 6=∅andC ∈ CM(LX_×LX_{), then} (a)Cis said to be semi-continuous from below if it satisfies: (c.v) For all semilattice from below{χi_}

i∈I ⊂LX and for allχ∈LX, the equality Inf

i∈IC(χ

i_{, χ) =C(Sup} i∈I

χi_{, χ) holds, where Sup} i∈I

χi _∈LX _{is defined as}

(Sup

i∈Iχ

i_{)(x) = Sup} i∈I χ

i_{(x), for allx∈X.}

(b)C is said to besemi-continuous from above if it satisfies: (c.vi) For all semilattice from above {χi_}

i∈I ⊂ LX\LX0 and for all χ ∈

LX_\LX

0, the equality Sup

i∈I

C(χi_{, χ) =C(Inf} i∈Iχ

i_{, χ) holds, where Inf} i∈Iχ

i _∈LX

is defined as (Inf

i∈Iχ

i_{)(x) = Inf} i∈Iχ

i_{(x) for allx∈X.}

Remark 3.1. Notice that, in section (b) of the previous definition, it is necessary to consider the AIFS be non-L-normal. Indeed, let X be an in-finite universe and let PF(X) be the family of all finite subsets of X. We

consider the semilattice from above{χA_}

A∈PF(X)⊂L

X _{such that for each}

A ∈ PF(X), χA(x) = 0L if x ∈ A, and χA(x) = 1L if x /∈ A. Thus, Inf

A∈PF(X)

χA_=χ0L _{and, ifC ∈ CM(L}X_×LX_{), it isC( Inf} A∈PF(X)

χA_{, χ}0L_{) = 1,}

nevertheless, asχA_∈LX

0 for allA∈ PF(X), Sup A∈PF(X)

C(χA_{, χ}0L) = 0.

The set of all semi-continuous from below contradiction measures will be denoted by CMsc(LX ×LX), and the set of all semi-continuous from

above contradiction measures byCMsc(LX_×LX_).

It is easy to prove that both (c.v) and (c.vi) imply (c.iv).

4. Some families of contradiction measures

Proposition 4.1. For any strong fuzzy negationN and for any order auto-morphism of unit intervalg, functionsCN,CgandCE defined in equation (1)

satisfy thatCN,Cg,CE∈ CMsc(LX×LX)butCN,Cg,CE∈ CM/ sc(LX×LX).

Proposition 4.2. Given p >0, for eachβ ∈[0,1]let us consider the seg-ment Lβ =

n

(α1, α2)∈[0,1]2|α1∈[0,1], α2= (α1+βp+)(1p−β)

o

. Let CL p,CpU : LX_×LX _{→[0,1]be defined for eachχ}A_{= (µ}

A, νA), χB = (µB, νB)∈LX

by

CL

p(χA, χB) =

(

0, ifχA_∈LX

0 orχB∈LX0 Min(1−β,1−β0_{), if Sup}

x∈X

χAB_(x)∈L

β& Sup x∈X

χBA_(x)∈L β0

CU

p(χA, χB) =

(

0, ifχA_∈LX

0 orχB∈LX0 Max(1−β,1−β0_{), if Inf}

x∈Xχ

AB_(x)∈L

β& Inf x∈Xχ

BA_(x)∈L β0

Then, for allp >0,CL

p ∈ CMsc(LX×LX)butCpL∈ CM/ sc(LX×LX), and CU

p ∈ CMsc(LX×LX)butCpU ∈ CM/ sc(LX×LX)(see Figure 3).

c

(0,0)

(1,1)

0IL

1IL

c

b

b

'

1-b

'

1 - b

(-p,0)

.

L

b

L

_b

_'

c

(0,0)

(1,1)

0IL

1IL .

c

.

b

b

'

1-b

'

1 - b

(-p,0)

L

_b

L

_b

_'

Fig. 3. Geometrical interpretation of measuresCpL andCpU.

Proposition 4.3. For eachp >0, the functionsC∨

p,Cp:LX×LX→[0,1],

defined for eachχA_{= (µ}

A, νA), χB= (µB, νB)∈LX by

Cp∨(χA, χB) = Min



 pxInf∈XνA(x)

p+ Sup

x∈XµB(x)

, pInf

x∈XνB(x)

p+ Sup

x∈XµA(x)

 

Cp(χA, χB) = Min

µ Inf

x∈X

pνA(x)

p+µB(x),xInf∈X

pνB(x)

p+µA(x)

¶ ,

satisfy thatC∨

Proposition 4.4. For each p >0, the functionsC∧

p,C˜p:LX×LX→[0,1],

defined for eachχA_{= (µ}

A, νA), χB= (µB, νB)∈LX by

Cp∧(χA, χB) =

    

0, if χA_∈LX

0 orχB ∈LX0

Max µ _pSup

x∈XνA(x)

p+ Inf x∈XµB(x)

,

pSup x∈XνB(x)

p+ Inf x∈XµA(x)

¶

, if χA_{, χ}B_∈LX_\LX

0

˜

Cp(χA, χB) =

  

0, ifχA_∈LX

0 or χB∈LX0

Max µ

Sup

x∈X pνA(x)

p+µB(x),Sup_x∈X

pνB(x)

p+µA(x) ¶

, ifχA_{, χ}B _∈LX_\LX

0 ,

satisfy thatC∧

p,C˜p∈ CMsc(LX×LX), butCp∧,C˜p∈ CM/ sc(LX×LX).

5. Conclusions

In the framework of AIFS, contradiction is a fact that must be considered and controlled, being necessary to measure how contradictory two sets are, in order to limit the degree accepted in any specific problem. This paper proposes a general model establishing the minimum requirements a function must satisfy for suitably measuring contradiction between two sets. Axioms presented take in account different kinds of continuity. Furthermore some families of functions are studied in accordance with this point of view.

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