Call for Papers for the 27th International Conference on Intuitionistic Fuzzy Sets is now open!
Conference: 5–6 July 2024, Burgas, Bulgaria • EXTENDED DEADLINE for submissions: 15 APRIL 2024.
Conference: 5–6 July 2024, Burgas, Bulgaria • EXTENDED DEADLINE for submissions: 15 APRIL 2024.
Issue:Approximations of crisp set and intuitionistic fuzzy set based on intuitionistic fuzzy normal subgroup: Difference between revisions
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| 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. |
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