Intuitionistic fuzzy sets

Intuitionistic fuzzy sets (IFSs) are an extension of the concept of Lotfi Zadeh's fuzzy sets, introduced originally in 1983 by the Bulgarian mathematician Krassimir Atanassov.

While in the case of fuzzy sets, a membership function $μ$ defines the level of membership of an element to a (fuzzy) set, the intuitionistic fuzzy set extends the fuzzy set by the introduction of a second function $ν$ which defines the level of non-membership of the element to the (intuitionistic fuzzy) set. Since both the membership and the non-membership function can be evaluated in the $[0, 1]$ interval, but do not necessarily sum up to 1, the complement to 1 is defined as degree of uncertainty or hesitation margin.

In this way, IFSs render a better account of uncertainty than fuzzy sets.

Formal definition

The formal definition of an intuitionistic fuzzy set is as follows.

Let $A$ be a set in the universe $U$ . Then for all elements $x$ from the universe, we call intuitionistic fuzzy set the set

A = {⟨ x, μ_{A} (x), ν_{A} (x) ⟩ | x \in E},

where $μ_{A} (x) : A \to [0, 1], ν_{A} (x) : A \to [0, 1]$ are called, respectively, the membership function and the non-membership function, and $μ_{A} (x) + ν_{A} (x) \in [0, 1]$ .

In the case when $ν_{A} (x) = 1 - μ_{A} (x)$ , the IFS is "flattened" to a fuzzy set.

Relations, operations, operators

Extensions of IFS

History of development

References

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