Abstract
Here, necessary corrections on the proof the Theorem 1 of Xu (J Intell Fuzzy Syst 33(3): 1563-1575, 2017) are stated in brief. Throughout, we use the same notations and equation numbers as in Xu.
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.
(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).
(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).
We take the four parts μ
A
(x), ν
A
(x),
In the paper [5], the proof of the third step is as follows.
3) For ∀A, B, C ∈ IFSs (X), we have
Thus, DIFSs (A, C) ≤ DIFSs (A, B) + DIFSs (B, C), which indicates that DIFSs satisfies (MD3).
However, the conclusion DIFSs (A, C) ≤ DIFSs (A, B) + DIFSs (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 2DIFSs (A, B) can be viewed as the Euclidean distance between two points
Since the distance measure 2DIFSs (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
Footnotes
Acknowledgment
The author would like to thank for a grant from the Ningxia Natural Science Foundation (No.2018AAC03253), the First-Class Disciplines Foundation of Ningxia (No.NXYLXK2017B09), the key project of North Minzu University (No. ZDZX201801, ZDZX201804), the National Natural Science Foundation of China (No. 61662001,11761002).
