Comment on “Improvement of the distance between intuitionistic fuzzy sets and its applications”

Intuitionistic fuzzy sets(IFSs) were proposed by Atanassov [1] as a generalization of the fuzzy sets. As the most interesting topics in IFSs theory, distance measures are involved in fuzzy decision making, patter recognition, fuzzy reasoning, etc, [2–6].

In 2017, a measuring distance between intuitionistic fuzzy sets, proposed by Xu [5], was successfully applied into pattern recognition problems and medical diagnosis. However, there is a small mistake about the proof of the Theorem 1 in Xu [5]. In order to show the detailed correction instructions, the definitions involved in the paper [5] are as follows.

Definition 1. A metric distance D in a non-empty set X is a real value function D : X × X → [0, + ∞), which satisfies the following conditions, ∀x, y, z  ∈  X:

(MD1) D (x, y) =0 if and only if x = y;

(MD2) D (x, y) = D (y, x);

(MD3) D (x, y) + D (y, z) ≥ D (x, z).

Definition 2. [7] Let D be a mapping: IFSs (X) × IFSs (X) → [0, 1]. For ∀A, B, C∈ IFSs(X), D (A, B) is a distance measure between IFSs A and B, if D satisfies the following properties:

(DP1) 0 ≤ D (A, B) ≤1;

(DP2) D (A, B) =0 if and only if A = B;

(DP3) D (A, B) = D (B, A);

(DP4) If A ⊆ B ⊆ C, then D (A, C) ≥ D (A, B), D (A, C) ≥D (B, C).

Definition 3. [5] Let A={〈x, μ_A (x) , v_A (x) 〉|x ∈ X} be a IFSs in X={x}, then the assignments of the hesitancy degree π_A (x) to membership degree μ_A (x) and non-membership degree ν_A (x) are defined as

We take the four parts μ_A (x), ν_A (x), AssignAπμ(x) and AssignAπν(x) into account the distances between IFSs, thereby a new distance measure, denoted as D_IFSs, is defined.

Definition 4. [5] Let A = {〈x, μ_A (x) , v_A (x) 〉|x ∈ X} , B = {〈x, μ_B (x) , v_B (x) 〉|x ∈ X} be two IFSs in X={x}, then the distance measure between A and B is defined as

Theorem 1. [5] Let A={〈x, μ_A (x) , v_A (x) 〉|x ∈ X}, B= {〈x, μ_B (x) , v_B (x) 〉|x ∈ X} be two IFSs in X={x}, then D_IFSs (A, B) is a distance measure satisfying the Definition 1 and Definition 2.

In the paper [5], the proof of the third step is as follows.

3) For ∀A, B, C ∈ IFSs (X), we have

Thus, D_IFSs (A, C) ≤ D_IFSs (A, B) + D_IFSs (B, C), which indicates that D_IFSs satisfies (MD3).

However, the conclusion D_IFSs (A, C) ≤ D_IFSs (A, B) + D_IFSs (B, C) is derived from the formula (3), which is a wrong logical reasoning. Where, it should be noted that the formula (3) is correct. In fact, from the formula (3) and the property of inequality, we can obtained

While, from the Definition 4, we have

Therefore,

However, based on the formula (5), it is not obtained

The proper proof of the third step is as follows.

3) According to the Definition 4, the distance measure 2D_IFSs (A, B) can be viewed as the Euclidean distance between two points (μA,νA,AssignAπμ,AssignAπν) and (μB,νB,AssignBπμ,AssignBπν) in a four-dimensional real number space. For ∀A, B, C ∈ IFSs (X), there are three real points.

Since the distance measure 2D_IFSs (A, B) is a metric distance, based on the third condition (MD3) of the Definition 1 (The properties of triangular inequalities for Euclidean distance), we have