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Soft Computing

, Volume 22, Issue 6, pp 2079–2079 | Cite as

Correction to: Redefinition of the concept of fuzzy set based on vague partition from the perspective of axiomatization

  • Xiaodong Pan
  • Yang Xu
Correction
  • 234 Downloads

1 Correction to: Soft Comput  https://doi.org/10.1007/s00500-017-2855-5

I recently found that several errors occur in the statement of Definition 5.2 in Section 5 in the paper “Redefinition of the concept of fuzzy set based on vague partition from the perspective of axiomatization”.

As it had been pointed out at the end of Section 4 of this paper that “the set of vague attribute values is defined as a free algebra on the elementary set of vague attribute values”, and fuzzy sets are mathematical formulation for vague attribute values, hence, the set of fuzzy sets in U can be seen as freely generated by a vague partition of U.

Based on this consideration, Definition 5.2 of this paper can be corrected as follows:

Definition 5.2

Let \(U = [a, b] \subset {\mathbb {R}}\) and \({\widetilde{U}} = \{\mu _{A_{1}}(x), \ldots , \mu _{A_{n}}(x)\}, n \in {\mathbb {N}}^{+},\) a vague partition of U. The set \({\mathscr {F}}({\widetilde{U}})\) of fuzzy sets in U with respect to \({\widetilde{U}}\) consists of the following elements:
  1. (1)

    if there exists \(i \in {\overline{n}}\) such that \(\mu _{A}(x) = \mu _{A_{i}}(x)\) for all \(x \in U\), then \(A = \{(x, \mu _{A}(x)) \mid x \in U\} \in {\mathscr {F}}({\widetilde{U}})\);

     
  2. (2)

    if \(\mu _{A}(x) = {\overline{\mu }}(x) = 1\) for all \(x \in U\), then \(A = \{(x, \mu _{A}(x)) \mid x \in U\} \in {\mathscr {F}}({\widetilde{U}})\);

     
  3. (3)

    if \(\mu _{A}(x) = \underline{\mu }(x) = 0\) for all \(x \in U\), then \(A = \{(x, \mu _{A}(x)) \mid x \in U\} \in {\mathscr {F}}({\widetilde{U}})\);

     
  4. (4)

    if \(A = \{(x, \mu _{A}(x)) \mid x \in U\} \in {\mathscr {F}}({\widetilde{U}})\) and \(r \in {\mathbb {Q}}^{+}\), then \(A^{r} = \{(x, (\mu _{A}(x))^{r}) \mid x \in U\} \in {\mathscr {F}}({\widetilde{U}})\);

     
  5. (5)

    if \(A = \{(x, \mu _{A}(x)) \mid x \in U\} \in {\mathscr {F}}({\widetilde{U}})\), and N is a strong negation on [0, 1], then \(A^{N} = \{(x, (\mu _{A}(x))^{N}) \mid x \in U\} \in {\mathscr {F}}({\widetilde{U}})\);

     
  6. (6)

    if \(A = \{(x, \mu _{A}(x)) \mid x \in U\}, B = \{(x, \mu _{B}(x)) \mid x \in U\} \in {\mathscr {F}}({\widetilde{U}})\), and \(\otimes \) is a triangular norm, then \(A \cap _{\otimes } B = \{(x, \mu _{A}(x) \otimes \mu _{B}(x)) \mid x \in U\} \in {\mathscr {F}}({\widetilde{U}})\);

     
  7. (7)

    if \(A = \{(x, \mu _{A}(x)) \mid x \in U\}, B = \{(x, \mu _{B}(x)) \mid x \in U\} \in {\mathscr {F}}({\widetilde{U}})\), and \(\oplus \) is a triangular conorm, then \(A \cup _{\oplus } B = \{(x, \mu _{A}(x) \oplus \mu _{B}(x)) \mid x \in U\} \in {\mathscr {F}}({\widetilde{U}})\);

     
  8. (8)

    \({\mathscr {F}}({\widetilde{U}})\) not include other elements.

    In fact, \({\mathscr {F}}({\widetilde{U}})\) can be considered as a function space based on \({\widetilde{U}}\).

     

We apologize to the readers for any inconvenience these errors might have caused.

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of MathematicsSouthwest Jiaotong UniversityChengduPeople’s Republic of China

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