Difference and symmetric difference for intuitionistic fuzzy sets

Fono et al. [10] determined some classes of difference and symmetric difference operations for fuzzy sets using fuzzy implication operators. Intuitionistic fuzzy sets are known to be generalizations of fuzzy sets. So, in this paper, we propose new difference and symmetric difference operations for intuitionistic fuzzy sets based on intuitionistic fuzzy R-implication operators and standard intuitionistic fuzzy negation operator. We establish that some common properties of the difference operations for fuzzy sets established earlier by Fono et al. in [10] and for crisp sets are preserved by the new obtained operations for intuitionistic fuzzy sets. We display a specific property satisfied by difference operation in crisp and fuzzy cases and violated in intuitionistic fuzzy case. The proposed difference and symmetric difference operations for intuitionistic fuzzy sets generalize the case for fuzzy sets. This strength provides a more dynamic perspective into the studies and applications of these operations.


Introduction
The framework of fuzzy set provides us with tools to handle problems in which the source of vagueness is the ambiguity in criteria of class membership rather than randomness [20]. In this framework, any element of a universal crisp set is allowed to belong to a subset partially with a membership grade usually between 0 and 1 assigned to it. Furthermore, the sum of the membership grade and non-membership grade of an element is always 1. But in reality, this case is not always true because there may be some hesitation degree [9] and this led to the introduction of intuitionistic fuzzy sets as generalization of fuzzy sets by Atanassov [1] in which the degree of hesitation is accounted for, so that the sum of the membership grade, non-membership grade of an element and its degree of hesitation is always 1. Throughout this paper, we consider that definition of an intuitionistic fuzzy set.
Since intuitionistic fuzzy set theory is a generalization of the fuzzy set theory, a rigorous study was needful to be able to establish workable results when concepts under crisp sets and fuzzy sets are transferred. A new set of definitions for set operations needed to be proposed for this field. Many standard operations (such as inclusion, intersection, union, complement, etc) [2,7,9,11,12] have been unanimously agreed upon to serve as usual operations on intuitionistic fuzzy sets. Meanwhile, the need to study these operations in a more mathematical framework which allows for generalization has motivated many scholars [3-6, 11-13, 17-19] among others to undertake studies in intuitionistic fuzzy operators and generators. Of these operators which are germane to establishing results in our current research include intuitionistic fuzzy t-norms, t-conorms, R-implications, co-implications and negations. Cornelis et al. [5,6] and Atanassov [3] have established many results in the study of intuitionistic fuzzy implications, co-implications, negations and their properties. Some of these results have provided in great measure some required mathematical background for our current study.
Fono et al. [10] have proposed two classes of difference operations for fuzzy sets and two classes of symmetric difference for fuzzy sets using the fuzzy implication operators. They established that these difference and symmetric difference operations for fuzzy sets of type 1 and 2 preserve the classical properties of difference and symmetric difference operations for crisp sets. Inspired by their work on fuzzy sets, we introduce new definitions for difference and symmetric difference for intuitionistic fuzzy sets by means of intuitionistic fuzzy R-implications and we study their properties.
Huawen [15] defined three difference operations for intuitionistic fuzzy sets, one based on the intuitionistic fuzzy t-norm T M = (min, max) and the remaining based on any decreasing intuitionistic fuzzy generators as follows: For any two intuitionistic fuzzy sets A and B of X, where ϕ is any decreasing intuitionistic fuzzy generator such that ϕ(0) = 1. However, these definitions do not provide a sufficient endowment to explore the mathematical extensions of these operations to the more general cases that apply to any intuitionistic fuzzy t-norm. Huawen's difference operations preserve only two out of the four properties which are the minimal conditions (as we have established in the results of Proposition 2) to require for a difference operation on sets, fuzzy sets and intuitionistic fuzzy sets in general. It is worthy to note here that the Huawen's difference operation − 1 is trivially the generalization of the difference operation in the sense of crisp set given by A−B = A∩B c . As he noted, if we choose the generator ϕ to be the standard negator defined by ϕ(x) = 1 − x, then the difference operations − 2 and − 3 reduce to − 1 . The complement functions (which are special examples of any difference operation) constructed from the difference operations − 2 and − 3 in Equations (2) and (3) are the same as the intuitionistic fuzzy complementation and intuitionistic fuzzy pseudo-complementation respectively, obtained by Bustince, et al. [4]. Thus, we can refer to the difference operation − 3 as intuitionistic fuzzy pseudo-difference operation, which in general does not inherit the general properties of the difference − 2 .
It is also notable to remark here that, the intuitionistic fuzzy complementation associated to difference − 2 defined by Huawen [15] and Bustince, et al. [4] depends largely on the choice of the intuitionistic fuzzy generator. Thus, with different choice of intuitionistic fuzzy generator, the intuitionistic fuzzy complementation so defined may yield different result. Meanwhile, the intuitionistic fuzzy complementation associated to the difference operation we proposed, though by means of intuitionistic fuzzy R-implications, yet yields the same result for any choice of associated t-representable intuitionistic fuzzy t-norm. By these new difference and symmetric difference operations for intuitionistic fuzzy sets we have proposed, we are able to construct typical examples of intuitionistic fuzzy difference and symmetric difference associated to the three usual known of intuitionistic fuzzy t-norms (the minimum, product and Lukasiewicz). More explicit examples of these new operations can be constructed for other t-representable intuitionistic fuzzy t-norms. This possibility provides a more robust knowledge and insight into the study of these operations in general cases and their applications would be more enriched.
The rest of this paper is organized as follows. Section 2 recalls some preliminaries on fuzzy sets and intuitionistic fuzzy sets. It also recalls known and useful results on difference and symmetric difference of fuzzy sets established by Fono et al. [10]. Section 3 introduces difference and symmetric difference of intuitionistic fuzzy sets and establishes their properties. Section 4 gives some concluding remarks. An Appendix recalls some known results of the fuzzy case that we use.

Preliminaries
Throughout this paper, X shall denote a nonempty universal set, a t-norm and S a t-conorm.
In this Section, we introduce some basic definitions and provide some preliminary results needed in the rest of the paper. Some other useful notions and concepts on fuzzy sets are recalled in Appendix.

Intuitionistic fuzzy sets, intuitionistic fuzzy operators and operations
Here, we introduce the basic concepts of intuitionistic fuzzy sets, recall the definitions and examples of some intuitionistic fuzzy operators and operations ( [5,13,17]).
Definition 1 (Intuitionistic Fuzzy Set [7,9,17]). An intuitionistic fuzzy set D on X is defined by: where µ D (x), ν D (x) are the degrees of membership and non-membership of x in D, respectively.
2. An intuitionistic fuzzy t-conorm is a binary operation J : L * × L * −→ L * such that for any x ∈ L * , J (x, 0 L * ) = x and, J is commutative, monotone increasing and associative.
Deschrijver et al. and, Reseir and Bedregal [6,18] have shown that an involutive intuitionistic fuzzy negation, N , can be characterized by an involutive fuzzy negation by proving that, if the fuzzy negation N is involutive, then N (x) = (N (1 − x 2 ), 1 − N (x 1 )). An example of a strong (involutive) intuitionistic fuzzy negation is the standard negation N s on L * defined by N s (x 1 , x 2 ) = (x 2 , x 1 ).
We now recall useful classes of intuitionistic fuzzy t-norm and t-conorm and, their implications and co-implications.
We denote by IF-t-norm the intuitionistic fuzzy t-norm and, by IF-t-conorm the intuitionistic fuzzy t-conorm.
1. An intuitionistic fuzzy R-implication (for short, IF-R-implication) associated with an IF-t-norm, T = ( , S), is a mapping I T : L * ×L * −→ L * such that, for all x = (x 1 , x 2 ), y = (y 1 , y 2 ) ∈ L * , The following are examples of t-representable IF-t-norms and IF-t-conorms [13]. ii. T P = ( P , S P ) and J P = (S P , P ) are t-representable IF-t-norm and IF-t-conorm respectively associated to P and S P .
iii. T L = ( L , S L ) and J L = (S L , L ) are t-representable IF-t-norm and IF-t-conorm respectively associated to L and S L .
iv. Also, by verifying that l are t-representable IF-t-norm and IF-t-conorm respectively associated to l F and S l F for all l ∈ (0, 1) ∪ (1, +∞). and 2. IF-R-implication associated with T P = ( P , S P ) and IF-co-implication associated with J P = (S P , P ) are respectively given by: and 3. The IF-R-implication associated with T L = ( L , S L ) and the IF-co-implication associated with J L = (S L , L ) are respectively given by: 4. The IF-R-implication associated with T l F = ( l F , S l F ) and the IF-co-implication associated with J l F = (S l F , l F ) for all l ∈ (0, 1) ∪ (1, +∞) are respectively given by: We end this Subsection by recalling inclusion and some operations on intuitionistic fuzzy sets.
Definition 6 (Intuitionistic Fuzzy Operations [2,7,9,11,12]). Let A and B be any two intuitionistic fuzzy sets defined on X. Inclusion and the following operations are defined by associated membership and non-membership functions as follows: In the next Subsection, we recall the difference and symmetric difference operations for fuzzy sets, some examples and their properties as proposed by Fono et al. [10].

Difference and symmetric difference of fuzzy sets based on fuzzy implications
b. The fuzzy symmetric difference of type i ∈ {1, 2} associated to of M and N is the fuzzy set of X denoted by M i N and defined for all x ∈ X by: We recall the examples of these operations for fuzzy sets of type 1 and 2 associated to the usual three fuzzy t-norms in what follows.
(c) The difference operation associated with L is given by, for all x ∈ X and i ∈ {1, 2},

Examples of fuzzy symmetric difference operations
(a) The symmetric difference operation associated with M is given by, for all The symmetric difference operation associated with P is given by, for all (c) The symmetric difference operation associated with L is given by, for all x ∈ X and Fono et al. [10] have also proved that the difference and symmetric difference operations for fuzzy sets of type 1 and 2 associated to any continuous t-norm so defined preserve the properties of the classical difference and symmetric difference operation for crisp sets. We recall these results as follows: 2. Properties of fuzzy symmetric difference operation; The following result shows that, the fuzzy complement of fuzzy sets associated with any continuous t-norm so defined, preserve the property of the classical complement for crisp sets.
Corollary 1. Let be any continuous t-norm, A be a fuzzy set on X, and A c be the fuzzy complement of A associated with .
Since µ X (x) = 1 and I (1, a) = a (see Proposition 9 in the Appendix) for all a ∈ [0; 1], from Definition 7, In the following Section, we introduce new operations for intuitionistic fuzzy sets and establish some of their properties.
3 New operations for intuitionistic fuzzy sets: Difference and symmetric difference

Definitions and properties of difference operations
Let I T = ( 1 I T , 2 I T ) be an IF-R-implication operator. We define the negation of I T as N (I T ) = ( 2 I T , 1 I T ). In particular, using Lemma 1 we define the negation of IF-R-implication as ∀x = ( Definition 8. Let A, B be any two intuitionistic fuzzy sets defined on X. The intuitionistic fuzzy difference associated to T of A and B is the intuitionistic fuzzy set on X denoted by A − T B and defined by the membership and non-membership degrees as follows: For all x ∈ X, The following are typical examples of difference operations associated with the three usual and well-known T .
Example 4. For any intuitionistic fuzzy sets A and B defined on X, 2. The difference operation associated with T P is given by, for all x ∈ X 3. The difference operation associated with T L is given by, for all x ∈ X In the following results, we establish four classical properties for difference operation which are satisfied by the new intuitionistic fuzzy difference operation.
Proposition 2 (Properties of Intuitionistic Fuzzy Difference Operation). Let A, B, C be intuitionistic fuzzy sets on X. The following properties for intuitionistic fuzzy difference operations hold: For the non-membership degree, there are four possibilities:
The following result shows that, the intuitionistic fuzzy complement of fuzzy sets associated with a t-representable of an IF-t-norm T = ( , S) so defined, preserve the property of the classical complement for crisp sets. Proof. Let x ∈ X. Since (µ X (x), ν X (x)) = (1, 0), then from Definition 8, , (recalling Prop. 9(1)), From Definition 6, the result follows.
The following result establishes a property of the new difference operation.
Proposition 3. Let A and B be any intuitionistic fuzzy sets on X.

Then
∀x ∈ X and non-membership function defined by: for all x ∈ X, 2. If T is a Lukasiewicz IF-t-norm, then Proof.
2. If T is Lukasiewicz IF-t-norm, then T = T L = ( L , S L ). Since from the result in 1 above, we have the membership function µ (A − T B)∩ T (B − T A) (x) = 0, ∀x ∈ X, then from Equation (9) it suffices to prove that the non-membership function, for the first two cases in (9). From Equation (11), , we obtain by applying Proposition 9 and Example 8, If µ A (x) < µ B (x) and ν A (x) < ν B (x), then we have , we obtain by applying Proposition 9 and Example 8, (0, 1), for all x ∈ X. Hence result 2 is established.
This follows immediately from the third case in Equation (9).
2. Proposition 2 specifies conditions which are preserved by the intuitionistic fuzzy difference operation. These four conditions shall be referred to as the minimal conditions to require of difference operation on (even in crisp, fuzzy and intuitionistic) sets in general.
The following result gives a necessary and sufficient condition for difference of intuitionistic fuzzy sets to be a fuzzy set.
Proof. Let x ∈ X. Then from the Definition 8, if and only if Note that A − T B also becomes a fuzzy set if A ⊂ B, because in this case A − T B = ∅ (Proposition 2), I (µ A (x), µ B (x) = 1 and J S (ν A (x), ν B (x)) = 0. Furthermore, in the case where A − T B becomes a fuzzy set, we deduce from Proposition 4 that for This can be considered as fuzzy part of A − T B.
The following are typical applications of Proposition 4 to difference operators associated with the three usual and well-known T . 2. The difference operation associated with T P is given by, for all x ∈ X 3. The difference operation associated with T L is given by, for all x ∈ X

Notation 1. Let A and B be any fuzzy sets, A ≡ B if and only if for all
The following result shows that the intuitionistic fuzzy difference operator defined in Definition 8 associated with t-representable IF t-norm T = ( , S) is a generalization of fuzzy difference operator proposed by Fono et al. [10] associated with a t-norm if and only if the fuzzy t-norm and fuzzy t-conorm S are dual.

Proposition 5 (Generalization of Difference Operation for Fuzzy Sets). Let and S be any tnorm and t-conorm respectively, and T = ( , S) be a t-representable IF t-norm associated with any intuitionistic fuzzy set. and S are dual if and only if for any fuzzy sets A and B, A − T B is a fuzzy set and
Proof. Let x ∈ X, and A and B be any fuzzy sets.
a. Assume that and S are dual.
i. Let us show that A − T B is a fuzzy set. Since and S are dual, then From Proposition 10, I (µ A (x), µ B (x)) = 1 − J S (1 − µ A (x), 1 − µ B (x)) and from Proposition 4, the result follows.
ii. Now we shall show that, According to Fono and al. [10], µ A− B (x) = 1 − I (µ A (x), µ B (x)) and from Definition 8 Since and S are dual, the Proposition 10 shows that, ) and the result follows.
b. Assume now that A − T B is a fuzzy set and A − B ≡ A − T B. Let us show that and S are dual. We have, and (12) and (13) , ∀t ∈ [0; 1] and the result follows.
In the following, we define a new symmetric difference operation for intuitionistic fuzzy sets based on the IF-R-implication and IF-co-implication and we study its properties.

Definitions and properties of symmetric difference operations
The idea for the new definition is derived from the classical formula for symmetric difference and the operations of union and intersection alongside with the proposed difference for intuitionistic fuzzy sets in Section 3.
Definition 9. Let A, B be any two intuitionistic fuzzy sets defined on X. The intuitionistic fuzzy symmetric difference associated to T of A and B is the intuitionistic fuzzy set on X denoted by A T B and defined by the membership and non-membership degrees as follows: For all x ∈ X, In what follows, we establish some results showing that some properties of the classical set symmetric difference are preserved by this new proposed intuitionistic fuzzy symmetric difference operation.
Proposition 6 (Properties of Intuitionistic Fuzzy Symmetric Difference Operation). Let A, B be any intuitionistic fuzzy sets on X. The following properties for intuitionistic fuzzy symmetric difference operation hold: Proof.
1. The following are properties for fuzzy-R-implication, I and fuzzy co-implication, J S which we require here: a, c).
These can easily be verified. Now, we proceed to prove 1 and 2 consequently as follows: From Equation (14) and applying above properties of I and J S we have, for all x ∈ X

So we have
and applying Proposition 9 we have the following: So, result 1 is established.
2. By commutativity of Equations (15) and (16), result 2 follows, since Applying the above inequalities to the Equation (14), we get and the result follows. (14) we have, for all x ∈ X

By Equation
So the result is established.
The following are typical examples of symmetric difference operators associated with the three usual and well-known T .
Example 6. For any two intuitionistic fuzzy sets A and B defined on X, 1. The symmetric difference operator associated with T M is given by, for all x ∈ X 2. The symmetric difference operator associated with T P is given by, for all x ∈ X 3. The symmetric difference operator associated with T L is given by, for all x ∈ X ).
The following result shows that, the intuitionistic fuzzy symmetric difference operator in Definition 9 associated with t-representable IF t-norm T = ( , S) is a generalization of fuzzy symmetric difference operator proposed by Fono et al. [10] associated with a t-norm if and only if the fuzzy t-norm and fuzzy t-conorm S are dual.
Proposition 7 (Generalization of Symmetric Difference Operation for Fuzzy Sets). Let and S be any t-norm and t-conorm, respectively, and T = ( , S) be a t-representable IF t-norm associated with any intuitionistic fuzzy set. and S are dual if and only if for any fuzzy sets C and D, C T D is a fuzzy set and C D ≡ C T D.
Proof. Let x ∈ X, and C and D be any fuzzy sets.
a. Assume that and S are dual.
i. Let us show that C T D is a fuzzy set. Since C and D are fuzzy sets (1 − µ C (x) = ν C (x) and 1 − µ D (x) = ν D (x)) , and and S are dual, then from Proposition 10, ). (17) From Definition 9, the result follows.
ii. Now we shall show that, C D ≡ C T D. It is sufficient to prove that µ C D (x) = µ C T D (x). Definition 7 and Definition 9 show that, From Equation (17), the result follows.
b. Assume now that C T D is a fuzzy set and C D ≡ C T D. Let us show that and S are dual. We have, (18) and Since . From Equation (18) and (19), , ∀t ∈ [0; 1], and the result follows.
In the following Subsection, we investigate some properties of cardinality for intuitionistic fuzzy set difference and symmetric difference. For that, throughout this Subsection, the universal set X is finite.

Some cardinality properties of difference and symmetric difference for IFSs
We recall the definition and some results on intuitionistic fuzzy cardinality in what follows.
Definition 10 (Cardinality of Intuitionistic Fuzzy Set [19]). Let A be an intuitionistic fuzzy set on X. The cardinality of A denoted by Σ count(A) is given by One of the properties of this cardinality operation is given here [See Property of Σ count [19] ]: Let A and B be any two intuitionistic fuzzy sets on X. Then In what follows, we establish a cardinality property that is satisfied by the intuitionistic fuzzy difference and symmetric difference proposed.
Proposition 8. Let A, B, C be any intuitionistic fuzzy sets on X. The following property holds: Putting Equation (23) into (22) we obtain the required result.

Conclusion
In this study, we have proposed new difference and symmetric difference operations for intuitionistic fuzzy sets by means of intuitionistic fuzzy R-implications. We have also constructed some examples of difference and symmetric operations associated to the well-known intuitionistic fuzzy t-norms (minimum T M , product T P and Łukasiewicz T L ) and established conditions under which these operations yield the same results for fuzzy cases as obtained by Fono et al. [10]. We further established that the intuitionistic fuzzy difference operation preserves four properties out of five, which we referred to as the four minimal conditions to require of a difference operation on sets in general (even in crisp, fuzzy and intuitionistic fuzzy cases). We investigated and established some sufficient conditions under which the fifth property is satisfied. Meanwhile, we established that the intuitionistic fuzzy symmetric difference operation proposed preserves the properties of symmetric difference operations for crisp sets and fuzzy sets. We established out of many, one cardinality property that is satisfied by these operations.
The results of Proposition 3 have shown that the property, (A − T B) ∩ T (B − T A) = ∅ do not hold true in general case for the difference operation for intuitionistic fuzzy sets proposed. The open problem will be to determine all intuitionistic fuzzy-t-norms under which the difference operation, so defined, preserves this property. We have not studied here, other cardinality properties and the cardinality-based measures of comparison for intuitionistic fuzzy sets by means of these new difference and symmetric difference operations for intuitionistic fuzzy sets proposed. This area is opened for further research studies.
The following examples of fuzzy t-norms and fuzzy t-conorms belonging to a family called Frank t-norms and Frank t-conorms will commonly be referred to in this study (see [10,13]): where M , P , L are the minimum, product and Lukasiewicz fuzzy t-norms, respectively. The where S M , S P , S L are the maximum, probabilistic sum and Lukasiewicz fuzzy t-conorms, respectively (see [9,10,13,16,17]). .
The following are examples of fuzzy R-implications and fuzzy co-implications associated with Frank t-norms and Frank t-conorms respectively. Example 8. [10,13,17]: for all a, b ∈ [0, 1]: 1. Fuzzy R-implication and fuzzy co-implication associated with M and S M are respectively given by and 2. Fuzzy R-implication and fuzzy co-implication associated with P and S P are respectively given by and 3. Fuzzy R-implication and fuzzy co-implication associated with L and S L are respectively given by and 4. Fuzzy R-implication and fuzzy co-implication associated with l F and S l F for all l ∈ (0, 1)∪ (1, +∞) are respectively given by ii) Intersection: A ∩ B is defined by: µ A∩B (x) = µ A (x) ∧ µ B (x), ∀ x ∈ X; iii) Union: Where ∨ and ∧ are max and min respectively.