Intuitionistic fuzzy sets and other fuzzy sets extensions representable by them

1 Introduction

Aristotle made the first steps to the establishment of the mathematical logic by formulating the concepts “sentence” and “predicate”; the fundamental logical functions as conjunction, disjunction and negation, the logical quantifiers, modal operators, and many others. He has the first who justifies the need for axioms and presented the first examples of such. For 23 centuries one of his proposed axioms – the law of excluded middle – has been among the main tools for proving mathematical assertions.

In the beginning of 20-th century, logical paradoxes appeared, which put in doubt the very foundations of mathematics. In the first decade of the century, David Hilbert and Bertrand Russel proposed ideas how the crisis in mathematics might be overcome, but with his theorem from the beginning of the 1930s Gödel showed the inapplicability of such ideas.

In 1912, Luitzen Brouwer proposed a new idea, called by him “intuitionism” which most generally may be stated as refusal to work with infinite sets and the law of excluded middle (see [9–12]).

In 1926, Jan Łukasiewicz for the first time proposed the sentences and predicates to be evaluated not by the two values 0 (“false”) and 1 (“true”), as has been done since the times of Aristotle, but to add an additional value ¹/₂ (“uncertainty”). This revolutionary step led to the emergence of a completely new type of logic. It is a serious argument supporting Brouwer’s intuitionism. In 1956, Łukasiewicz generalized the proposed by him three-valued logic to n-valued (many-valued logic).

The next step in the development of this idea was made by Lotfi Zadeh in 1965 by the introduction of the concept “fuzzy set” [16]. Just several years after the introduction of fuzzy sets, they became the object of further generalizations: L-fuzzy sets, rough sets, etc.

In 1983, the intuitionistic fuzzy sets were defined, in which for the first time two degrees were proposed of membership, or validity, or correctness (μ) and of non-membership, or non-validity, or non-correctness, etc (ν) [1]. Behind this definition is clearly seen Brouwer’s idea for intuitionism  because every sentence, predicate, object, etc is evaluated not only as true (μ) or false (ν), but also by the degree of indeterminacy π. This was the reason the new sets were called intuitionistic fuzzy sets. Their definition does not preclude the possibility to define over such sets operations of the classical logic - for instance, negation and implication, however, it also provides the opportunity to define a wide class of nonclassical operations: negation, implication, conjunction, disjunction, etc.

In the recent years in the literature devoted to fuzzy sets there appeared definitions of objects named with the word “intuitionistic” but without the word “fuzzy”. This is incorrect because the reader is left with the impression that these objects are from the area of Brouwer’s intuitionism, but not from the area of Zadeh’s fuzzy sets.

After their introduction, by the end of 1980s intuitionistic fuzzy sets became an object of generalizations – intuitionistic L-fuzzy, interval valued intuitionistic fuzzy sets, intuitionistic fuzzy sets of second (and more generally n-th)-type, temporal intuitionistic fuzzy sets.

In the present paper we will discuss one of the generalization of the IFS – Interval valued intuitionistic fuzzy sets, and we will show how we can use them to represent some of the modifications of the notion intuitionistic fuzzy sets that have appeared in the last twenty years.

2 Intuitionistic fuzzy sets and interval valued intuitionistic fuzzy sets

Let us have a fixed universe E and its subset A. The set A * = { 〈 x , μ A ( x ) , ν A ( x ) 〉 | x ∈ E } ,(1) where 0 ≤ μ A ( x ) + ν A ( x ) ≤ 1(2) is called IFS and functions μ _A  : E → [0, 1] and ν _A  : E → [0, 1] represent respectively the degree of membership (validity, etc.) and non-membership (non-validity, etc.) of element x ∈ E to a fixed set A ⊆ E (see, also [2, 6]). Now, we can define also function π _A  : E → [0, 1] by means of π ( x ) = 1 - μ ( x ) - ν ( x )(3) and it corresponds to the degree of indeterminacy (uncertainty, etc.).

For brevity, we shall write below A instead of A^*, whenever this is possible.

First, we mention that in [4] some geometrical interpretations are discussed. The second of them has the form from Fig. 1. The set of the points of the interpretation triangle from Fig. 1 can be written as L = { 〈 p , q 〉 | p , q , p + q ∈ [ 0 , 1 ] } .(4)

OPEN IN VIEWER

Fig.1

Geometrical interpretation of IFS element.

An IVIFS A over E is an object of the form: A = { 〈 x , M A ( x ) , N A ( x ) 〉 ∣ x ∈ E } ,(5) where

M A ( x ) = [ inf M A ( x ) , sup M A ( x ) ] ⊆ [ 0 , 1 ] ,(6) N A ( x ) = [ inf N A ( x ) , sup N A ( x ) ] ⊆ [ 0 , 1 ](7) are intervals and for all x ∈ E: sup M A ( x ) + sup N A ( x ) ≤ 1(8) (see [3, 8]).

Obviously, this definition is analogous to the definition of IFS.

IVIFSs have geometrical interpretations similar to, but more complex than these of the IFSs. For example, the analogous of the geometrical interpretation from Fig. 1 is shown on Fig. 2.

OPEN IN VIEWER

Fig.2

Geometrical interpretation of IVIFS element.

Following [7], for the arbitrary element x ∈ E with degrees a and b (i.e., a, b, a + b ∈ [0, 1]), for the triple 〈x, a, b〉 we call that it is nesting in the IVIFS A and denote 〈 x , a , b 〉 ɛ A if and only if ( iff ) a ∈ M A ( x ) , b ∈ N A ( x ) .(9)

We will say that the IFS C = { 〈 x , μ C ( x ) , ν C ( x ) 〉 | x ∈ E }(10) is nesting in the IVIFS A iff for each x ∈ E, the triple 〈x, μ _C  (x) , ν _C  (x) 〉ɛA.

Now, let us define for the fixed above IVIFS A, A _ = { 〈 x , inf M A ( x ) , sup N A ( x ) 〉 | x ∈ E } ,(11) A ¯ = { 〈 x , sup M A ( x ) , inf N A ( x ) 〉 | x ∈ E } .(12)

Obviously, for each IVIFS A, A _ ⊆ A ¯ .(13)

Moreover, sets A _ and A ¯ are IFSs and they are determined bijectively from A. In [7], the following two assertions are proved.

Theorem 1. Let A be an IVIFS and B be an IFS. A _ ⊆ B ⊆ A ¯ iff B is nesting in A .(14) We call the two IFSs A _ and A ¯ lower and upper boundary IFSs of the IVIFS A.∥Theorem 2. For every two IFSs B and C over universe E, such that for each x ∈ E sup M B ( x ) + sup N C ≤ 1(15) and C ⊆ B ,(16) there exists a unique IVIFS A with upper and lower boundary IFSs A ¯ = B and A _ = C .∥From both theorems it follows that each IVIFS can be represented by a pair of two IFSs.

In [14], the concept of an Inconsistent Intuitionistic Fuzzy Set (IIFS) is introduced, as follows A i = { 〈 x , μ A i ( x ) , ν A i ( x ) , ι A i ( x ) 〉 | x ∈ E } ,(17) where 0 ≤ μ A i ( x ) + ν A i ( x ) + ι A i ( x ) ≤ 1(18) is called IFS and functions μ A i : E → [ 0 , 1 ] and ν A i : E → [ 0 , 1 ] represent, respectively, the degree of membership (validity, etc.) and non-membership (non-validity, etc.) of element x ∈ E to a fixed set A ⊆ E .

We can define again function π _A  : E → [0, 1] by means of π A i ( x ) = 1 - μ A i ( x ) - ν A i ( x ) - ι A i ( x )(19) and it corresponds to degree of indeterminacy (uncertainty, etc.).

Now, we will discuss four IVIFS-interpretations of a fixed IIFS A ⁱ .

Let: inf M A ( x ) = μ A i ( x ) ,(20) inf N A ( x ) = ν A i ( x ) ,(21) sup M A ( x ) = μ A i ( x ) μ A i ( x ) + ν A i ( x ) ( 1 - ι A i ( x ) ) ,(22) sup N A ( x ) = ν A i ( x ) μ A i ( x ) + ν A i ( x ) ( 1 - ι A i ( x ) ) .(23)

Obviously, sup M A ( x ) + sup N A ( x ) = 1 - ι A ( x ) ≤ 1 .(24)

Therefore, the first IVIFS-interpretation of the IIFS A ⁱ is:

A = { 〈 x , [ μ A i ( x ) , μ A i ( x ) μ A i ( x ) + ν A i ( x ) ( 1 - ι A i ( x ) ) ] , [ ν A i ( x ) , ν A i ( x ) μ A i ( x ) + ν A i ( x ) ( 1 - ι A i ( x ) ) ] 〉 ∣ x ∈ E } .(25)

Let: inf M A ( x ) = μ A i ( x ) ,(26) inf N A ( x ) = ν A i ( x ) ,(27) sup M A ( x ) = μ A i ( x ) μ A i ( x ) + ν A i ( x ) ( 1 - π A i ( x ) ) ,(28) sup N A ( x ) = ν A i ( x ) μ A i ( x ) + ν A i ( x ) ( 1 - π A i ( x ) ) .(29)

Obviously, sup M A ( x ) + sup N A ( x ) = 1 - π A ( x ) ≤ 1 .(30)

Therefore, the second IVIFS-interpretation of the IIFS A ⁱ is:

A = { 〈 x , [ μ A i ( x ) , μ A i ( x ) μ A i ( x ) + ν A i ( x ) ( 1 - ι A i ( x ) ) ] , [ ν A i ( x ) , ν A i ( x ) μ A i ( x ) + ν A i ( x ) ( 1 - ι A i ( x ) ) ] 〉 ∣ x ∈ E } .(31)

Let: inf M A ( x ) = μ A i ( x ) ,(32) inf N A ( x ) = ν A i ( x ) ,(33) sup M A ( x ) = μ A i ( x ) + ι A i ( x ) ,(34) sup N A ( x ) = ν A i ( x ) + π A i ( x ) .(35)

Obviously, sup M A ( x ) + sup N A ( x ) = 1 .(36)

Therefore, the third IVIFS-interpretation of the IIFS A ⁱ is:

A = { 〈 x , [ μ A i ( x ) , μ A i ( x ) + ι A i ( x ) ] , = [ ν A i ( x ) , ν A i ( x ) + π A i ( x ) ] 〉 ∣ x ∈ E } .(37)

Let: inf M A ( x ) = μ A i ( x ) ,(38) inf N A ( x ) = ν A i ( x ) ,(39) sup M A ( x ) = μ A i ( x ) + π A i ( x ) ,(40) sup N A ( x ) = ν A i ( x ) + ι A i ( x ) .(41)

Obviously, sup M A ( x ) + sup N A ( x ) = 1 .(42)

Therefore, the fourth IVIFS-interpretation of the IIFS A ⁱ is:

A = { 〈 x , [ μ A i ( x ) , μ A i ( x ) + π A i ( x ) ] , [ ν A i ( x ) , ν A i ( x ) + ι A i ( x ) ] 〉 ∣ x ∈ E } .(43)

The concept of a Picture Fuzzy Set was introduced by B. C. Cuong in 2013 (see [13]). A PFS A^* is an object of the following form: A * = { 〈 x , μ A ( x ) , η A ( x ) , ν A ( x ) 〉 | x ∈ E }(44) where the functions μ _A  : E → [0, 1] and η _A  (x) : E → [0, 1] and ν _A  : E → [0, 1] are called “degree of positive membership”, “degree of neutral membership” and “degree of negative membership” of x in A^*, respectively, and for all x ∈ E, μ A ( x ) + η A ( x ) + ν A ( x ) ≤ 1 .(45) The degree of “refusal of membership” is thus understood as: r _A  (x) =1 - μ _A  (x) - η _A  (x) - ν _A  (x) .

Obviously, this concept coincides totally with IIFS and therefore, it has the above four IVIFS-interpretations.

In [15], the concept of a Neutrosophic Fuzzy Set (NFS) is introduced, as follows A n = { 〈 x , μ A n ( x ) , ν A n ( x ) , π A n ( x ) 〉 | x ∈ E } ,(46) where μ A n ( x ) , ν A n ( x ) , π A n ( x ) ∈ [ 0 , 1 ] and have the same sense as IFS.

Let sup y ∈ E μ A n ( y ) + sup y ∈ E ν A n ( y ) + sup y ∈ E π A n ( y ) ≠ 0 .(47) Then we define: μ A i ( x ) = μ A n ( x ) sup y ∈ E μ A n ( y ) + sup y ∈ E ν A n ( y ) + sup y ∈ E π A n ( y ) ,(48) ν A i ( x ) = ν A n ( x ) sup y ∈ E μ A n ( y ) + sup y ∈ E ν A n ( y ) + sup y ∈ E π A n ( y ) ,(49) π A i ( x ) = π A n ( x ) sup y ∈ E μ A n ( y ) + sup y ∈ E ν A n ( y ) + sup y ∈ E π A n ( y ) ,(50) ι A i ( x ) = 1 - μ A i ( x ) - ν A i ( x ) - π A i ( x ) .(51)

If sup y ∈ E μ A n ( y ) + sup y ∈ E ν A n ( y ) + sup y ∈ E π A n ( y ) = 0 ,(52) then we define: μ A i ( x ) = 0 ,(53) ν A i ( x ) = 0 ,(54) π A i ( x ) = 0 ,(55) ι A i ( x ) = 1 .(56)

Therefore, the NFS can be represented by an IIFS. Using the above four IVIFS-interpretations of IIFS, we can construct four IVIFS-interpretations of a NFS.

If for the element x ∈ E : μ A n ( x ) + ν A n ( x ) + π A n ( x ) ≠ 0 , then we define: μ A ( x ) = μ A n ( x ) μ A n ( x ) + ν A n ( x ) + π A n ( x ) ,(57) ν A ( x ) = ν A n ( x ) μ A n ( x ) + ν A n ( x ) + π A n ( x ) ,(58) π A ( x ) = π A n ( x ) μ A n ( x ) + ν A n ( x ) + π A n ( x ) .(59)

If μ A n ( x ) + ν A n ( x ) + π A n ( x ) = 0 , then we define: μ A ( x ) = 0 ,(60) ν A ( x ) = 0 ,(61) π A ( x ) = 1 .(62)

Therefore, μ _A  (x) + ν _A  (x) + π _A  (x) =1, i.e. A = { 〈 x , μ A ( x ) , ν A ( x ) 〉 | x ∈ E }(63) is an IFS.

Therefore, the NFS can be represented by an IFS.

Here as an illustration we provide few simple example of the different types and their corresponding representation for one element sets. Let us consider the IIFS (it can also be viewed as PFS): A i = { 〈 x , 0.3 , 0.4 , 0.2 〉 | x ∈ E } ,(64) Its first corresponding representations by IVIFS is: A = { 〈 x , [ 3 10 , 24 70 ] , [ 4 10 , 32 70 ] 〉 ∣ x ∈ E }(65)

Let A n = { 〈 x , 0.7 , 0.7 , 0.9 〉 | x ∈ E } ,(66) be a NFS. Then its corresponding IFS is: A = { 〈 x , 7 23 , 7 23 〉 | x ∈ E } ,(67)

Consider the IVIFS from (65). Then its upper and lower boundary IFSs are: A _ = { 〈 x , 3 10 , 32 70 〉 | x ∈ E } ,(68) A ¯ = { 〈 x , 24 70 , 4 10 〉 | x ∈ E } .(69)

As it has become clear from the constructions provided above, any Inconsistent intuitionistic fuzzy set, and Picture fuzzy set, and Neutrosophic fuzzy set are representable by an Interval-valued intuitionistic fuzzy set, which itself is representable by an ordered pair of standard intuitionistic fuzzy sets. Therefore, the assertions of certain authors that IIFS, PFS and NFS are extensions of IFS is incorrect. It is evident that all these extensions of fuzzy sets are equipollent with IFS.

In future study, we will show that other fuzzy set extensions are also equipollent with IFS.