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B. K. Tripathy (VIT University, India) and K. R. Arun (VIT University, India)

Source Title: Handbook of Research on Generalized and Hybrid Set Structures and Applications for Soft Computing

Copyright: © 2016
|Pages: 21
DOI: 10.4018/978-1-4666-9798-0.ch005

Chapter Preview

TopIn this section, we shall introduce several definitions including that of soft set and operations on them are to be presented.

**Definition (Soft set):**Let U is a classical set of elements, the pair (U, E) is called the universe and E be a set of parameters often regarded as a soft universe. Members of the universe and the parameter set are generally denoted by x and e respectively. A soft set over the soft universe (U, E) is denoted by (F, A), where*F*:*A*→*P*(*U*), where A is a subset of E and P (U) is the power set of U which comprises of all the subsets of U.

Parameter Reduction: Reducing the number of parameters using any techniques and core parameter is identified for easy manipulation.

Data Clustering: The process of putting a given data set into groups of similar elements is called data clustering.

Soft Set: soft set was introduced by Molodtsov(1999) . Let U is a classical set of elements, the pair (U, E) is called the universe and E be a set of parameters often regarded as a soft universe. Members of the universe and the parameter set are generally denoted by x and e respectively. A soft set over the soft universe (U, E) is denoted by (F, A), where F : A ? P ( U ), where A is a subset of E and P (U) is the power set of U which comprises of all the subsets of U.

Rough Set: Rough set theory was initiated by Pawlak (1982). Let U be a universe and R be an equivalence relation over U. This equivalence relation decomposes U into disjoint equivalence classes. We denote the equivalence class of an element x with respect to R by [ x ] R , which is defined as [ x ] R = { y | yRx }. Then for any , we associate two crisp sets and called the lower and upper approximations of X with respect to R respectively and are defined as, = { x ? U: X} and = { x ? U: [ x ] R n X }.

Fuzzy Set: Fuzzy set was introduced by Zadeh (1965). Fuzzy set is the extension of crisp set. In fuzzy, each and every element has the degrees of membership value which lies between [0, 1]. A fuzzy set is the pair ( S , µ) where S is a set and µ : S ? [0,1].

Intuitionistic Fuzzy Set: An (IFS) A over E, is characterised by two functions µ A and ? A called the membership and non-membership function of A respectively such that µ A :U ? [0, 1] and ? A :U ? [0, 1]. For any x ? U , we have 0 = µ A ( x ) + ? A ( x ) = 1. The hesitation function of A is denoted by p A and for any x ? U , is given by p A ( x ) = 1 – µ A ( x ) – ? A ( x ). It is the indeterministic part of x.

Incomplete Soft Set: Missing of some data in the soft set due to the incomplete database exist in literature.

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