Trigonometric-based operators over intuitionistic fuzzy sets

: In the present paper we introduce two new operators over intuitionistic fuzzy sets and study their properties.

Definition 1 (cf. [1,3,5]). Let X be a universe set, A ⊂ X. Then an intuitionistic fuzzy set generated by the set A is an object of the form: where µ A : X → [0, 1] and ν A : X → [0, 1] are mappings, such that for any x ∈ X, Let us further denote the class of all IFSs over the same universe X, by IFS(X). If S is a mapping S : IFS(X) → IFS(X), we call S an operator defined over IFS (X).
Some examples of operators previously defined include but are not limited to (see [5]): where α ∈ [0, 1]; In what follows we propose two new operators based on the trigonometric functions sin(t) and cos(t).

The proposed operators
First we start with the following simple observations: • In the interval [0, π 2 ], the function cos(t) is non-negative and strictly decreasing.
• Both function are bounded from below by 0 and from above by 1 on the interval [0, π 2 ].
Another key point to note is that due to (1. Now we are ready to define our two operators. Let A * be the IFS defined by (1.1).

Definition 2.
We define the operator Z cos : IFS(X) → IFS(X) as follows: Definition 3. We define the operator Z sin : IFS(X) → IFS(X) as follows: We will show that Z cos (A * ) ∈ IFS(X) and Z sin (A * ) ∈ IFS(X), i.e. the operators are correctly defined. It is clear that We have (due to (2.1)) It is easy to see that Thus, we obtain that We can easily see that: and hence (2.4) is an IFS. In order to see how these operators behave we consider the following intuitionistic fuzzy set We can see that the new operators act similarly to the classical negation in the sense that if µ B (x) > ν B (x), we have µ Z (x) < ν Z (x), and vice versa.

Remark 1.
If we denote Z cos (A * ) = ¬ (Z cos (A * )) , and Z sin (A * ) = ¬ (Z sin (A * )) , we obtain another two operators Z cos and Z sin over the intuitionistic fuzzy sets.

Proposition 2.
The following relationships are fulfilled: Proof. We will only go through the first equality since the others are checked in the same manner.
For the left-hand side we obtain: For the right-hand side: Thus both sides yield the same IFS.

Conclusion and an open problem
In the present work we introduced two new operators over intuitionistic fuzzy sets based on the trigonometric functions sin and cos . An interesting Open problem is the following: Can the operator X a,b,c,d,e,f , defined by (see [5])