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# Issue:Approximations of crisp set and intuitionistic fuzzy set based on intuitionistic fuzzy normal subgroup

 shortcut http://ifigenia.org/wiki/issue:nifs/23/4/91-105
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
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 $\displaystyle{ A=(\mu_A,\nu_A) }$ of G. For each α, β ∈ [0, 1] such that α + β ≤ 1, the set $\displaystyle{ U (A, \alpha, \beta) = U (\mu_A , \alpha) \cap U (\nu_A, \beta) }$ is a congruence relation on G, where $\displaystyle{ U (\mu_A , \alpha) = \{ (x,y) \in G \times G: \mu_A(xy^{-1}) \geq \alpha \} }$ and $\displaystyle{ U (\nu_A , \beta) = \{ (x,y) \in G \times G: \nu_A(xy^{-1}) \leq \beta \} }$. 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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