Sequences, Nets, and Filters of Fuzzy Soft Multi Sets in Fuzzy Soft Multi Topological Spaces

2.1 FUZZY SET THEORY

Zadeh (1965) introduced the concept of fuzzy sets as a new mathematical tool for modeling uncertainty and it made its own place in decision making problems. The word fuzzy means “vagueness”. Fuzzy sets have been introduced by Zadeh (1965) as an extension of the classical characteristic function notation of set. Let A be a crisp set defined over the universe X. Then for any element x in X, either x is a member of A or not. In fuzzy set theory, this property is generalized. Zadeh extended the range of the membership function to the closed interval [0, 1] from two element set {0, 1}.

Definition ( Zadeh,1965): If X is a collection of objects then a fuzzy set A in X is a set of ordered pairs: A={(x, μ_A(x)) /x ∈ X} where μ_A(x) is called the membership function of x in A which maps X to the membership space [0,1]. i.e., μ_A: X →[0,1] .

The words like “young”, “tall”, “good” or “high” are fuzzy concepts. There is no single quantitative value which defines the term “young”, “good” or “high”.

Let A and B be fuzzy sets on a universal set X, with the grade of membership of x in A and B denoted by μ_A and μ_B respectively. Zadeh (1965) defined the following relations and operations.

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  A = B ⇔ μ_A(x) = μ_B(x) for all x in X;

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  A ⊆ B ⇔ μ_A(x) ≤ μ_B(x) for all x in X;

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  μ_A’ (x) = 1-μ_A(x) for all x in X; A’ is the complement of A.

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  μ_A∪B (x) = Max{μ_A(x), μ_B (x) } for all x in X;

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  μ_A∩B (x) = Min{μ_A(x), μ_B (x) } for all x in X;