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Example 1.1. Let F be the set of all fuzzy subsets of a set Ω.I ff : Ω → [0, 1] then we define A =(f ,1− f), i.e. ν A = 1 − μ A. Example 1.2. Let (Ω, S) be a measurable space, S a σ-algebra, F the family of all pairs such that μ A : Ω → [0, 1], ν A : Ω → [0, 1] are measurable. Then F is closed under the operations ⊕, ⊙, ¬. Example 1.3. Let (Ω, T) be a topological space, F the family of all pairs such that μ A : Ω → [0, 1], ν A : Ω → [0, 1] are continuous. Then F is closed under the operations⊕, ⊙, ¬. Remark. Of course, in any case A ⊕ B, A ⊙ B, ¬A are IF-sets, if A, B are IF-sets. E.g.

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