**16-17 May 2019 • Sofia, Bulgaria**

**Submission:**21 February 2019 •

**Notification:**11 March 2019 •

**Final Version:**1 April 2019

# Difference between revisions of "Intuitionistic fuzzy sets"

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where <math>0 \leq \mu_A(x) + \nu_A(x) \leq 1</math>. We will call the set '''''A*''''' '''intuitionistic fuzzy set''' (IFS). | where <math>0 \leq \mu_A(x) + \nu_A(x) \leq 1</math>. We will call the set '''''A*''''' '''intuitionistic fuzzy set''' (IFS). | ||

− | In the publications on IFS authors mainly deal with the concept of intuitionistic fuzzy set '''''A*''''' rather then with fixed set '''''A'''''. This is why, for the sake of simplicity, major publications presenting the very definition of the concept often use notation '''''A''''' instead of '''''A*'''''.<ref>Paper "Intuitionistic Fuzzy sets", Krassimir T. Atanassov, [[Fuzzy Sets and Systems]], North-Holland, Volume 20 (1986), pages 87-96, ISSN 0165-0114</ref><ref>Book [[Intuitionistic Fuzzy | + | In the publications on IFS authors mainly deal with the concept of intuitionistic fuzzy set '''''A*''''' rather then with fixed set '''''A'''''. This is why, for the sake of simplicity, major publications presenting the very definition of the concept often use notation '''''A''''' instead of '''''A*'''''.<ref>Paper "Intuitionistic Fuzzy sets", Krassimir T. Atanassov, [[Fuzzy Sets and Systems]], North-Holland, Volume 20 (1986), pages 87-96, ISSN 0165-0114</ref><ref>Book [[Intuitionistic Fuzzy Sets: Theory and Applications|"Intuitionistic Fuzzy Sets"]], Krassimir T. Atanassov, Series "Studies in Fuzziness and Soft Computing", Volume 35, Springer Physica-Verlag, 1999, ISBN 3-7908-1228-5</ref> Mathematically, a more precise definition of the IFS is the following: |

<div align="center"><math>A^* = \lbrace \langle x, \mu_A(x), \nu_A(x) \rangle \ | \ x \in E \ \& \ 0 \leq \mu_A(x) + \nu_A(x) \leq 1 \rbrace</math></div> | <div align="center"><math>A^* = \lbrace \langle x, \mu_A(x), \nu_A(x) \rangle \ | \ x \in E \ \& \ 0 \leq \mu_A(x) + \nu_A(x) \leq 1 \rbrace</math></div> | ||

but it is also more complex one and never used, as of 2008.<ref>"25 years intuitionistic fuzzy sets, or: The most significant results and mistakes of mine", Krassimir Atanassov, [[International Workshop on Intuitionistic Fuzzy Sets and Generalized Nets/2008|Int. Workshop on IFS and GN]], 17 Oct. 2008 ([[Media:7IWIFSGN-Atanassov.pdf|presentation]], publication in press)</ref> | but it is also more complex one and never used, as of 2008.<ref>"25 years intuitionistic fuzzy sets, or: The most significant results and mistakes of mine", Krassimir Atanassov, [[International Workshop on Intuitionistic Fuzzy Sets and Generalized Nets/2008|Int. Workshop on IFS and GN]], 17 Oct. 2008 ([[Media:7IWIFSGN-Atanassov.pdf|presentation]], publication in press)</ref> |

## Revision as of 12:09, 10 November 2008

**Intuitionistic fuzzy sets** are sets whose elements have degrees of membership and non-membership. Intuitionistic fuzzy sets have been introduced by Krassimir Atanassov (1983) as an extension of Lotfi Zadeh's notion of fuzzy set, which itself extends the classical notion of a set.

- In classical set theory, the membership of elements in a set is assessed in binary terms according to a bivalent condition — an element either belongs or does not belong to the set.
- As an extension, fuzzy set theory permits the gradual assessment of the membership of elements in a set; this is described with the aid of a membership function valued in the real unit interval [0, 1].
- The theory of intuitionistic fuzzy sets further extends both concepts by allowing the assessment of the elements by two functions: for membership and for non-membership, which belong to the real unit interval [0, 1] and whose sum belongs to the same interval, as well.

Intuitionistic fuzzy sets generalize fuzzy sets, since the indicator functions of fuzzy sets are special cases of the membership and non-membership functions and of intuitionistic fuzzy sets, in the case when the strict equality exists: , i.e. the non-membership function fully complements the membership function to 1, not leaving room for any uncertainty.

## Contents

## Formal definition

Let us have a fixed universe * E*. Let

*be a subset of*

**A***. Let us construct the set*

**E**where . We will call the set **A*****intuitionistic fuzzy set** (IFS).

In the publications on IFS authors mainly deal with the concept of intuitionistic fuzzy set * A** rather then with fixed set

*. This is why, for the sake of simplicity, major publications presenting the very definition of the concept often use notation*

**A***instead of*

**A***.*

**A***^{[1]}

^{[2]}Mathematically, a more precise definition of the IFS is the following:

but it is also more complex one and never used, as of 2008.^{[3]}

Functions and represent degree of membership (validity, etc.) and non-membership (non-validity, etc.). Also defined is function through , corresponding to the degree of uncertainty (indeterminacy, etc.)

Obviously, for every ordinary fuzzy set : for each and these sets have the form

## Properties of IFS

## Geometrical interpretations

## Relations with other concepts

## Applications of IFS

## History of IFS

## References

- ↑ Paper "Intuitionistic Fuzzy sets", Krassimir T. Atanassov, Fuzzy Sets and Systems, North-Holland, Volume 20 (1986), pages 87-96, ISSN 0165-0114
- ↑ Book "Intuitionistic Fuzzy Sets", Krassimir T. Atanassov, Series "Studies in Fuzziness and Soft Computing", Volume 35, Springer Physica-Verlag, 1999, ISBN 3-7908-1228-5
- ↑ "25 years intuitionistic fuzzy sets, or: The most significant results and mistakes of mine", Krassimir Atanassov, Int. Workshop on IFS and GN, 17 Oct. 2008 (presentation, publication in press)