On the Most Extended Modal Operator of First Type over Interval-Valued Intuitionistic Fuzzy Sets

Abstract

E→P([0,1]) represent the set of degrees of membership (validity, etc.) and the set of degrees of non-membership (non-validity, etc.) of element x∈E x∈E to a fixed set A⊆E A⊆E , where P(Z)={Y|Y⊆Z} P(Z)={Y|Y⊆Z} for an arbitrary set Z. Z. Obviously, both intervals have the representation: MA(x)=[infMA(x),supMA(x)], NA(x)=[infNA(x),supNA(x)]. [...]from (5) it is clear that it will be enough to check that X=a2supMA(x)+b2(1−supMA(x)−c2supNA(x)) +d2supNA(x)+e2(1−f2supMA(x)−supNA(x)) =(a2−b2−e2 f2)supMA(x)+(d2−e2−b2 c2)supNA(x)+b2+e2≤1. [...]the definition of the IVIFS is correct. ☐ All of the operators described above can be represented by the operator Xa1b1c1d1e1f1a2b2c2d2e2f2 Xa1b1c1d1e1f1a2b2c2d2e2f2at suitably chosen values of its parameters. Proof. (c) Let a1,b1,c1,d1,e1,f1,a2,b2,c2,d2,e2,f2∈[0,1] a1,b1,c1,d1,e1,f1,a2,b2,c2,d2,e2,f2∈[0,1]satisfy (2)–(5) , and let A and B be fixed IVIFSs. [...]the infMA infMA -component of IVIFS Y is higher than or equal to the infMA infMA -component of IVIFS Z. In the same manner, it can be checked that the same inequality is valid for the supMA supMA -components of these IVIFSs. In Advances in Data Analysis with Computational Intelligence Methods; Studies in Computational Intelligence; Gawęda, A., Kacprzyk, J., Rutkowski, L., Yen, G., Eds.; Springer: Cham, Switzerland, 2018; Volume 738, pp. 265–274.

1. Introduction

Intuitionistic fuzzy sets (IFSs; see [1,2,3,4,5]) were introduced in 1983 as an extension of the fuzzy sets defined by Lotfi Zadeh (4.2.1921–6.9.2017) in [6]. In recent years, the IFSs have also been extended: intuitionistic L -fuzzy sets [7], IFSs of second [8] and nth [9,10,11,12] types, temporal IFSs [4,5,13], multidimensional IFSs [5,14], and others. Interval-valued intuitionistic fuzzy sets (IVIFSs) are the most detailed described extension of IFSs. They appeared in 1988, when Georgi Gargov (7.4.1947–9.11.1996) and the author read Gorzalczany’s paper [15] on the interval-valued fuzzy set (IVFS). The idea of IVIFS was announced in [16,17] and extended in [4,18], where the proof that IFSs and IVIFSs are equipollent generalizations of the notion of the fuzzy set is given.

Over IVIFS, many (more than the ones over IFSs) relations, operations, and operators are defined. Here, similar to the IFS case, the standard modal operators ☐ and ◇ have analogues, but their extensions—the intuitionistic fuzzy extended modal operators of the first type—already have two different forms. In the IFS case, there is an operator that includes as a partial case all other extended modal operators. In the present paper, we construct a similar operator for the case of IVIFSs and study its properties.

2. Preliminaries

Let us have a fixed universe E and its subset A. The set

A={⟨x,_MA(x),_NA(x)⟩∣x∈E},

where _MA(x)⊂[0,1]and_NA(x)⊂[0,1] _MA(x)⊂[0,1]and_NA(x)⊂[0,1] are closed intervals and for all x∈E x∈E:

sup_MA(x)+sup_NA(x)≤1

is called IVIFS, and functions _MA:E→P([0,1]) _MA:E→P([0,1]) and _NA:E→P([0,1]) _NA:E→P([0,1]) represent the set of degrees of membership (validity, etc.) and the set of degrees of non-membership (non-validity, etc.) of element x∈E x∈E to a fixed set A⊆E A⊆E , where P(Z)={Y|Y⊆Z} P(Z)={Y|Y⊆Z} for an arbitrary set Z. Z.

Obviously, both intervals have the representation:

_MA(x)=[inf_MA(x),sup_MA(x)],

_NA(x)=[inf_NA(x),sup_NA(x)].

Therefore, when

inf_MA(x)=sup_MA(x)=_μA(x)andinf_NA(x)=sup_NA(x)=_νA(x),

the IVIFS A is transformed to an IFS.

We must mention that in [19,20] the second geometrical interpretation of the IFSs is given (see Figure 1).

Figure 1. The second geometrical interpretation of an intuitionistic fuzzy set (IFS).

IVIFSs have geometrical interpretations similar to, but more complex than, those of the IFSs. For example, the analogue of the geometrical interpretation from Figure 1 is shown in Figure 2.

Figure 2. The second geometrical interpretation of an interval-valued intuitionistic fuzzy set (IVIFS).

Obviously, each IVFS A can be represented by an IVIFS as

A={⟨x,_MA(x),_NA(x)⟩∣x∈E}

={⟨x,_MA(x),[1−sup_MA(x),1−inf_MA(x)]⟩∣x∈E}.

The geometrical interpretation of the IVFS A is shown in Figure 3. It has the form of a section lying on the triangle’s hypotenuse.

Figure 3. The second geometrical interpretation of an IVFS.

Modal-type operators are defined similarly to those defined for IFSs, but here they have two forms: shorter and longer. The shorter form is:

☐A={⟨x,_MA(x),[inf_NA(x),1−sup_MA(x)]⟩∣x∈E},◇A={⟨x,[inf_MA(x),1−sup_NA(x)],_NA(x)⟩∣x∈E},_Dα(A)={⟨x,[inf_MA(x),sup_MA(x)+α(1−sup_MA(x)−sup_NA(x))],[inf_NA(x),sup_NA(x)+(1−α)(1−sup_MA(x)−sup_NA(x))]⟩∣x∈E},_Fα,β(A)={⟨x,[inf_MA(x),sup_MA(x)+α(1−sup_MA(x)−sup_NA(x))],[inf_NA(x),sup_NA(x)+β(1−sup_MA(x)−sup_NA(x))]⟩∣x∈E},forα+β≤1,_Gα,β(A)={⟨x,[αinf_MA(x),αsup_MA(x)],[βinf_NA(x),βsup_NA(x)]⟩∣x∈E},_Hα,β(A)={⟨x,[αinfM(x),αsup_MA(x)],[inf_NA(x),sup_NA(x)+β(1−sup_MA(x)−sup_NA(x))]⟩∣x∈E},_Hα,β∗(A)={⟨x,[αinf_MA(x),αsup_MA(x)],[inf_NA(x),sup_NA(x)+β(1−αsup_MA(x)−sup_NA(x))]⟩∣x∈E},_Jα,β(A)={⟨x,[inf_MA(x),sup_MA(x)+α(1−sup_MA(x)−sup_NA(x))],[βinf_NA(x),βsup_NA(x)]⟩∣x∈E},_Jα,β∗(A)={⟨x,[inf_MA(x),sup_MA(x)+α(1−sup_MA(x)−βsup_NA(x))],[βinf_NA(x),β.sup_NA(x)]⟩∣x∈E},

where α,β∈[0,1]. α,β∈[0,1].

Obviously, as in the case of IFSs, the operator _{Dα Dα} is an extension of the intuitionistic fuzzy forms of (standard) modal logic operators ☐ and ◇, and it is a partial case of _{Fα,β Fα,β}.

The longer form of these operators (operators ☐, ◇, and D do not have two forms—only the one above) is (see [4]):

_F¯αγβδ(A)={⟨x,[inf_MA(x)+α(1−sup_MA(x)−sup_NA(x)),sup_MA(x)+β(1−sup_MA(x)−sup_NA(x))],[inf_NA(x)+γ(1−sup_MA(x)−sup_NA(x)),sup_NA(x)+δ(1−sup_MA(x)−sup_NA(x))]⟩∣x∈E}whereβ+δ≤1,_G¯αγβδ(A)={⟨x,[αinf_MA(x),βsup_MA(x)],[γinf_NA(x),δsup_NA(x)]⟩∣x∈E},_H¯αγβδ(A)={⟨x,[αinf_MA(x),βsup_MA(x)],[inf_NA(x)+γ(1−sup_MA(x)−sup_NA(x)),sup_NA(x)+δ(1−sup_MA(x)−sup_NA(x))]⟩∣x∈E},_{H¯αγβδ∗}(A)={⟨x,[αinf_MA(x),βsup_MA(x)],[inf_NA(x)+γ(1−βsup_MA(x)−sup_NA(x)),sup_NA(x)+δ(1−βsup_MA(x)−sup_NA(x))]⟩∣x∈E},_J¯αγβδ={⟨x,[inf_MA(x)+α(1−sup_MA(x)−sup_NA(x)),sup_MA(x)+β(1−sup_MA(x)−sup_NA(x))],[γinf_NA(x),δsup_NA(x)]⟩∣x∈E},_{J¯αγβδ∗}(A)={⟨x,[inf_MA(x)+α(1−δsup_MA(x)−sup_NA(x)),sup_MA(x)+β(1−sup_MA(x)−δsup_NA(x))],[γ.inf_NA(x),δ.sup_NA(x)]⟩∣x∈E},

where α,β,γ,δ∈[0,1] α,β,γ,δ∈[0,1] such that α≤β α≤β and γ≤δ γ≤δ.

Figure 4 shows to which region of the triangle the element x∈E x∈E (represented by the small rectangular region in the triangle) will be transformed by the operators F,G,..., F,G,...,irrespective of whether they have two or four indices.

Figure 4. Region of transformation by the application of the operators.

3. Operator X X

Now, we introduce the new operator

_{Xa1b1c1d1e1f1a2b2c2d2e2f2}(A)

={⟨x,[_a1inf_MA(x)+_b1(1−inf_MA(x)−_c1inf_NA(x)),

_a2sup_MA(x)+_b2(1−sup_MA(x)−_c2sup_NA(x))],

[_d1inf_NA(x)+_e1(1−_f1inf_MA(x)−inf_NA(x)),

_d2sup_NA(x)+_e2(1−_f2sup_MA(x)−sup_NA(x))]⟩|x∈E},

where _a1,_b1,_c1,_d1,_e1,_f1,_a2,_b2,_c2,_d2,_e2,_f2∈[0,1] _a1,_b1,_c1,_d1,_e1,_f1,_a2,_b2,_c2,_d2,_e2,_f2∈[0,1] , the following three conditions are valid for i=1,2 i=1,2:

_ai+_ei−_{ei fi}≤1,

_bi+_di−_{bi ci}≤1,

_bi+_ei≤1,

and

_a1≤_a2,_b1≤_b2,_c1≤_c2,_d1≤_d2,_e1≤_e2,_f1≤_f2.

Theorem 1.

For every IVIFS A and for every _a1,_b1,_c1,_d1,_e1,_f1,_a2,_b2,_c2,_d2,_e2,_{f2 a1},_b1,_c1,_d1,_e1,_f1,_a2,_b2,_c2,_d2,_e2,_f2 ∈[0,1] ∈[0,1] that satisfy (2)–(5), _{Xa1b1c1d1e1f1a2b2c2d2e2f2}(A) _{Xa1b1c1d1e1f1a2b2c2d2e2f2}(A) is an IVIFS.

Proof.

Let _a1,_b1,_c1,_d1,_e1,_f1,_a2,_b2,_c2,_d2,_e2,_f2∈[0,1] _a1,_b1,_c1,_d1,_e1,_f1,_a2,_b2,_c2,_d2,_e2,_f2∈[0,1]satisfy (2)–(5) and let A be a fixed IVIFS. Then, from (5) it follows that

_a1inf_MA(x)+_b1(1−inf_MA(x)−_c1inf_NA(x))

≤_a2sup_MA(x)+_b2(1−sup_MA(x)−_c2sup_NA(x))

and

_d1inf_NA(x)+_e1(1−_f1inf_MA(x)−inf_NA(x))

≤_d2sup_NA(x)+_e2(1−_f2sup_MA(x)−sup_NA(x)).

Now, from (5) it is clear that it will be enough to check that

X=_a2sup_MA(x)+_b2(1−sup_MA(x)−_c2sup_NA(x))

+_d2sup_NA(x)+_e2(1−_f2sup_MA(x)−sup_NA(x))

=(_a2−_b2−_{e2 f2})sup_MA(x)+(_d2−_e2−_{b2 c2})sup_NA(x)+_b2+_e2≤1.

In fact, from (2),

_a2−_b2−_{e2 f2}≤1−_b2−_e2

and from (3):

_d2−_e2−_{b2 c2}≤1−_b2−_e2.

Then, from (1),

X≤(1−_b2−_e2)(sup_MA(x)+sup_NA(x))+_b2+_e2

≤1−_b2−_e2+_b2+_e2=1.

Finally, when sup_MA(x)=inf_NA(x)=0 sup_MA(x)=inf_NA(x)=0and from (4),

X=_b2(1−0−0)+_e2(1−0−0)=_b2+_e2≤1.

Therefore, the definition of the IVIFS is correct. ☐

All of the operators described above can be represented by the operator _{Xa1b1c1d1e1f1a2b2c2d2e2f2 Xa1b1c1d1e1f1a2b2c2d2e2f2}at suitably chosen values of its parameters. These representations are the following:

☐A=_{X10r110s110r2111}(A),◇A=_{X10r110s111110s2}(A),_Dα(A)=_{X10r11α110r211−α1}(A),_Fα,β(A)=_{X10r110s11α11β1}(A),_Gα,β(A)=_{Xα0r1β0s1α0r2β0s2}(A),_Hα,β(A)=_{Xα0r110s1α0r21β1}(A),_Hα,β∗(A)=_{Xα0r1α0s110r21βα}(A),_Jα,β(A)=_{X10r1β0s11α1β0s2}(A),_Jα,β∗(A)=_{X10r1β0s11αββ0s2}(A),_F¯αγβδ(A)=_{X1α11γ11β11δs1}(A),

_G¯αγβδ(A)=_{Xα0r1β0s1γ0r2δ0s2}(A),_H¯αγβδ(A)=_{Xα0r11γ1β0r21δ1}(A),_{H¯αγβδ∗}(A)=_{Xα0r11γ1β0r21δβ}(A),_J¯αγβδ(A)=_{X1α1γ0s11β1δ0s2}(A),_{J¯αγβδ∗}(A)=_{X1αδγ0s11βδδ0s2}(A),

where _r1,_r2,_s1,_{s2 r1},_r2,_s1,_s2 are arbitrary real numbers in the interval [0,1] [0,1].

Three of the operations, defined over two IVIFSs A and B, are the following:

¬A={⟨x,_NA(x),_MA(x)⟩∣x∈E},A∩B={⟨x,[min(inf_MA(x),inf_MB(x)),min(sup_MA(x),sup_MB(x))],[max(inf_NA(x),inf_NB(x)),max(sup_NA(x),sup_NB(x))]⟩∣x∈E},A∪B={⟨x,[max(inf_MA(x),inf_MB(x)),max(sup_MA(x),sup_MB(x))],[min(inf_NA(x),inf_NB(x)),min(sup_NA(x),sup_NB(x))]⟩∣x∈E}.

For any two IVIFSs A and B, the following relations hold:

A⊂Biff∀x∈E,inf_MA(x)≤inf_MB(x),inf_NA(x)≥inf_NB(x),sup_MA(x)≤sup_MB(x)andsup_NA(x)≥sup_NB(x)),A⊃BiffB⊂A,A=BiffA⊂BandB⊂A.

Theorem 2.

For every two IVIFSs A and B and for every _a1,_b1,_c1,_d1,_e1,_f1,_a2,_b2, _a1,_b1,_c1,_d1,_e1,_f1,_a2,_b2, _c2,_d2,_e2,_{f2 c2},_d2,_e2,_f2 ∈[0,1] ∈[0,1] that satisfy (2)–(5),

(a) ¬_{Xa1b1c1d1e1f1a2b2c2d2e2f2}(¬A)=_Xd,e,f,a,b,c(A), ¬_{Xa1b1c1d1e1f1a2b2c2d2e2f2}(¬A)=_Xd,e,f,a,b,c(A),

(b) _{Xa1b1c1d1e1f1a2b2c2d2e2f2}(A∩B) _{Xa1b1c1d1e1f1a2b2c2d2e2f2}(A∩B)

⊂_{Xa1b1c1d1e1f1a2b2c2d2e2f2}(A)∩_{Xa1b1c1d1e1f1a2b2c2d2e2f2}(B),

(c) _{Xa1b1c1d1e1f1a2b2c2d2e2f2}(A∪B) _{Xa1b1c1d1e1f1a2b2c2d2e2f2}(A∪B)

⊃_{Xa1b1c1d1e1f1a2b2c2d2e2f2}(A)∪_{Xa1b1c1d1e1f1a2b2c2d2e2f2}(B).

Proof.

(c) Let _a1,_b1,_c1,_d1,_e1,_f1,_a2,_b2,_c2,_d2,_e2,_f2∈[0,1] _a1,_b1,_c1,_d1,_e1,_f1,_a2,_b2,_c2,_d2,_e2,_f2∈[0,1]satisfy (2)–(5) , and let A and B be fixed IVIFSs. First, we obtain:

Y=_{Xa1b1c1d1e1f1a2b2c2d2e2f2}(A∪B)

=_{Xa1b1c1d1e1f1a2b2c2d2e2f2}({⟨x,[max(inf_MA(x),inf_MB(x)),

max(sup_MA(x),sup_MB(x))],

[min(inf_NA(x),inf_NB(x)),min(sup_NA(x),sup_NB(x))]⟩∣x∈E})

={⟨x,[_a1max(inf_MA(x),inf_MB(x))+_b1(1−max(inf_MA(x),inf_MB(x))

−_c1min(inf_NA(x),inf_NB(x))),_a2max(sup_MA(x)sup_MB(x))

+_b2(1−max(sup_MA(x)sup_MB(x))−_c2min(sup_NA(x),sup_NB(x)))],

[_d1min(inf_NA(x),inf_NB(x))+_e1(1−_f1max(inf_MA(x),inf_MB(x))

−min(inf_NA(x),inf_NB(x))),_d2min(sup_NA(x),sup_NB(x))

+_e2(1−_f2max(sup_MA(x)sup_MB(x))−min(sup_NA(x),sup_NB(x)))]⟩|x∈E}.

Second, we calculate:

Z=_{Xa1b1c1d1e1f1a2b2c2d2e2f2}(A)∪_{Xa1b1c1d1e1f1a2b2c2d2e2f2}(B)

={⟨x,[_a1inf_MA(x)+_b1(1−inf_MA(x)−_c1inf_NA(x)),

_a2sup_MA(x)+_b2(1−sup_MA(x)−_c2sup_NA(x))],

[_d1inf_NA(x)+_e1(1−_f1inf_MA(x)−inf_NA(x)),

_d2sup_NA(x)+_e2(1−_f2sup_MA(x)−sup_NA(x))]⟩|x∈E}

∪{⟨x,[_a1inf_MB(x)+_b1(1−inf_MB(x)−_c1inf_NB(x)),

_a2sup_MB(x)+_b2(1−sup_MB(x)−_c2sup_NB(x))],

[_d1inf_NB(x)+_e1(1−_f1inf_MB(x)−inf_NB(x)),

_d2sup_NB(x)+_e2(1−_f2sup_MB(x)−sup_NB(x))]⟩|x∈E}

={⟨x,[max(_a1inf_MA(x)+_b1(1−inf_MA(x)−_c1inf_NA(x)),

_a1inf_MB(x)+_b1(1−inf_MB(x)−_c1inf_NB(x))),

max(_a2sup_MA(x)+_b2(1−sup_MA(x)−_c2sup_NA(x)),

_a2sup_MB(x)+_b2(1−sup_MB(x)−_c2sup_NB(x)))],

[min(_d1inf_NA(x)+_e1(1−_f1inf_MA(x)−inf_NA(x)),

_d1inf_NB(x)+_e1(1−_f1inf_MB(x)−inf_NB(x))),

min(_d2sup_NA(x)+_e2(1−_f2sup_MA(x)−sup_NA(x)),

_d2sup_NB(x)+_e2(1−_f2sup_MB(x)−sup_NB(x)))]⟩∣x∈E}.

Let

P=_a1max(inf_MA(x),inf_MB(x))+_b1(1−max(inf_MA(x),inf_MB(x))

−_c1min(inf_NA(x),inf_NB(x)))−max(_a1inf_MA(x)+_b1(1−inf_MA(x)−_c1inf_NA(x)),

_a1inf_MB(x)+_b1(1−inf_MB(x)−_c1inf_NB(x)))

=_a1max(inf_MA(x),inf_MB(x))+_b1−_b1max(inf_MA(x),inf_MB(x))

−_{b1 c1}min(inf_NA(x),inf_NB(x)))−max((_a1−_b1)inf_MA(x)+_b1−_{b1 c1}inf_NA(x),

(_a1−_b1)inf_MB(x)+_b1−_{b1 c1}inf_NB(x))

=_a1max(inf_MA(x),inf_MB(x))−_b1max(inf_MA(x),inf_MB(x))

−_{b1 c1}min(inf_NA(x),inf_NB(x))−max((_a1−_b1)inf_MA(x)−_{b1 c1}inf_NA(x),

(_a1−_b1)inf_MB(x)−_{b1 c1}inf_NB(x)).

Let inf_MA(x)≥inf_MB(x) inf_MA(x)≥inf_MB(x). Then

P=(_a1−_b1)inf_MA(x)−_{b1 c1}min(inf_NA(x),inf_NB(x))−max((_a1−_b1)inf_MA(x)

−_{b1 c1}inf_NA(x),(_a1−_b1)inf_MB(x)−_{b1 c1}inf_NB(x)).

Let (_a1−_b1)inf_MA(x)−_{b1 c1}inf_NA(x)≥(_a1−_b1)inf_MB(x)−_{b1 c1}inf_NB(x). (_a1−_b1)inf_MA(x)−_{b1 c1}inf_NA(x)≥(_a1−_b1)inf_MB(x)−_{b1 c1}inf_NB(x).Then

P=(_a1−_b1)inf_MA(x)−_{b1 c1}min(inf_NA(x),inf_NB(x))−(_a1−_b1)inf_MA(x)

+_{b1 c1}inf_NA(x)

=_{b1 c1}inf_NA(x)−_{b1 c1}min(inf_NA(x),inf_NB(x))≥0.

If (_a1−_b1)inf_MA(x)−_{b1 c1}inf_NA(x)<(_a1−_b1)inf_MB(x)−_{b1 c1}inf_NB(x). (_a1−_b1)inf_MA(x)−_{b1 c1}inf_NA(x)<(_a1−_b1)inf_MB(x)−_{b1 c1}inf_NB(x).Then

P=(_a1−_b1)inf_MA(x)−_{b1 c1}min(inf_NA(x),inf_NB(x))−(_a1−_b1)inf_MB(x)

+_{b1 c1}inf_NB(x)).

=_{b1 c1}inf_NB(x)−_{b1 c1}min(inf_NA(x),inf_NB(x))≥0.

Therefore, the inf_MA inf_MA -component of IVIFS Y is higher than or equal to the inf_MA inf_MA -component of IVIFS Z. In the same manner, it can be checked that the same inequality is valid for the sup_MA sup_MA -components of these IVIFSs. On the other hand, we can check that that the inf_NA inf_NA - and sup_NA sup_NA -components of IVIFS Y are, respectively, lower than or equal to the inf_NA inf_NA and sup_NA sup_NA-components of IVIFS Z. Therefore, the inequality (c) is valid. ☐

4. Conclusions

In the near future, the author plans to study some other properties of the new operator _{Xa1b1c1d1e1f1a2b2c2d2e2f2 Xa1b1c1d1e1f1a2b2c2d2e2f2}.

In [21], it is shown that the IFSs are a suitable tool for the evaluation of data mining processes and objects. In the near future, we plan to discuss the possibilities of using IVIFSs as a similar tool.

Funding

This research was funded by the Bulgarian National Science Fund under Grant Ref. No. DN-02-10/2016.

Conflicts of Interest

The author declares no conflict of interest.

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