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Issue:Approximations of crisp set and intuitionistic fuzzy set based on intuitionistic fuzzy normal subgroup: Difference between revisions

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  | format          = PDF
  | size            = 157 Kb
  | size            = 157 Kb
  | abstract        = We consider a group <em>G</em>, with identity element <em>e</em>, as a universal set and assume that the knowledge about objects is restricted by an intuitionistic fuzzy normal subgroup <math>A=(\mu_A,\nu_A)</math> of G. For each α, β ∈ [0, 1] such that α + β ≤ 1, the set <math>U (A, \alpha, \beta) = U (\mu_A , \alpha) \cap U (\nu_A, \beta)</math> is a congruence relation on G, where <math>U (\mu_A , \alpha) = \{ (x,y) \in G \times G: \mu_A(xy^{-1} \geq \alpha \} and U (\nu_A , \beta) = \{ (x,y) \in G \times G: \nu_A(xy^{-1} \leq \beta \} </math>. In this paper, the notion of U (A, α, β)-lower and U (A, α, β)-upper approximation of a non-empty subset of G and an intuitionistic fuzzy set of G are introduced and some important properties of the above approximations are presented.
  | abstract        = We consider a group <em>G</em>, with identity element <em>e</em>, as a universal set and assume that the knowledge about objects is restricted by an intuitionistic fuzzy normal subgroup <math>A=(\mu_A,\nu_A)</math> of G. For each α, β ∈ [0, 1] such that α + β ≤ 1, the set <math>U (A, \alpha, \beta) = U (\mu_A , \alpha) \cap U (\nu_A, \beta)</math> is a congruence relation on G, where <math>U (\mu_A , \alpha) = \{ (x,y) \in G \times G: \mu_A(xy^{-1}) \geq \alpha \} </math> and <math>U (\nu_A , \beta) = \{ (x,y) \in G \times G: \nu_A(xy^{-1}) \leq \beta \} </math>. In this paper, the notion of U (A, α, β)-lower and U (A, α, β)-upper approximation of a non-empty subset of G and an intuitionistic fuzzy set of G are introduced and some important properties of the above approximations are presented.
  | keywords        = Rough set, Fuzzy set, Intuitionistic fuzzy set, Intuitionistic fuzzy normal subgroup, Congruence relation, Lower and Upper approximations of crisp and fuzzy sets.
  | keywords        = Rough set, Fuzzy set, Intuitionistic fuzzy set, Intuitionistic fuzzy normal subgroup, Congruence relation, Lower and Upper approximations of crisp and fuzzy sets.
  | ams            = 03E72, 06F35, 08A72, 03E99.
  | ams            = 03E72, 06F35, 08A72, 03E99.

Revision as of 02:34, 21 December 2017

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Title of paper: Approximations of crisp set and intuitionistic fuzzy set based on intuitionistic fuzzy normal subgroup
Author(s):
Prasenjit Mandal
Bhalukdungri Jr High School, Raigara, Purulia (W.B.), 723153, India
prasenjitmandal08@yahoo.com
A. S. Ranadive
Department of Pure and Applied Mathematics, Guru Ghasidas University, Bilaspur (C.G.), India
asranadive04@yahoo.co.in
Published in: "Notes on IFS", Volume 23, 2017, Number 4, pages 91—105
Download:  PDF (157 Kb  Kb, File info)
Abstract: We consider a group G, with identity element e, as a universal set and assume that the knowledge about objects is restricted by an intuitionistic fuzzy normal subgroup [math]\displaystyle{ A=(\mu_A,\nu_A) }[/math] of G. For each α, β ∈ [0, 1] such that α + β ≤ 1, the set [math]\displaystyle{ U (A, \alpha, \beta) = U (\mu_A , \alpha) \cap U (\nu_A, \beta) }[/math] is a congruence relation on G, where [math]\displaystyle{ U (\mu_A , \alpha) = \{ (x,y) \in G \times G: \mu_A(xy^{-1}) \geq \alpha \} }[/math] and [math]\displaystyle{ U (\nu_A , \beta) = \{ (x,y) \in G \times G: \nu_A(xy^{-1}) \leq \beta \} }[/math]. In this paper, the notion of U (A, α, β)-lower and U (A, α, β)-upper approximation of a non-empty subset of G and an intuitionistic fuzzy set of G are introduced and some important properties of the above approximations are presented.
Keywords: Rough set, Fuzzy set, Intuitionistic fuzzy set, Intuitionistic fuzzy normal subgroup, Congruence relation, Lower and Upper approximations of crisp and fuzzy sets.
AMS Classification: 03E72, 06F35, 08A72, 03E99.
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