Fuzzy approximation based on $ \tau- \mathfrak{K} $ fuzzy open (closed) sets

Priti; Alka Tripathi

doi:10.3934/mfc.2023010

Article Contents

2023, Volume 6, Issue 3: 558-572. Doi: 10.3934/mfc.2023010

This issue Previous Article The family of $ \lambda $-Bernstein-Durrmeyer operators based on certain parameters Next Article Starlike functions associated with $ \tanh z $ and Bernardi integral operator

Fuzzy approximation based on $ \tau- \mathfrak{K} $ fuzzy open (closed) sets

Priti^1,2, , and
Alka Tripathi^1,

1.
Department of Mathematics, Jaypee Institute of Information Technology, Noida, Uttar Pradesh India
2.
ABES Engineering College, Ghaziabad, Uttar Pradesh, India

^*Corresponding author: Priti
^*Corresponding author: Priti

Received: July 2022

Revised: January 2023

Early access: March 2, 2023

Published online: March 2, 2023

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Abstract

Rough set theory can be generalized by induced topology through equivalence relations. Motivated by the work of generalization of the rough set via topology, the concept and properties of $ \tau-\mathfrak{K} $-fuzzy open (closed) sets are proposed. Considering the $ \tau-\mathfrak{K} $-fuzzy open (closed) sets, we have obtained the $ \tau-\mathfrak{K} $-fuzzy lower and upper approximations and also proved their properties. $ \tau-\mathfrak{K} $-fuzzy open sets can be represented as $ \tau-\mathfrak{K} $-open sets by $ \alpha $ level sets. The properties of $ \tau-\mathfrak{K} $-fuzzy approximations and fuzzy rough approximations on the basis of binary fuzzy relation are compared. Finally, an example and the decision method's algorithm to illustrate the $ \tau-\mathfrak{K} $-fuzzy approximation-based approach to decision making are presented.

Keywords:

Mathematics Subject Classification: Primary: 54A05, 54A40; Secondary: 03E72.

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Table 1.

$ \mathcal{A}=\mathcal{P}(\mathcal{Z}_1) $	$ cl(\mathcal{A}) $	$ \tau_1-\mathfrak{K} $-FOS or not	$ \tau_1-\mathfrak{K} $-FCS	$ \underline{\mathfrak{K}}_{\mathfrak{K}}(\mathcal{A}) $	$ \overline{\mathfrak{K}}_{\mathfrak{K}}(\mathcal{A}) $
$ \big\{\frac{.4}{\mathfrak{u}_1}+\frac{0}{\mathfrak{u}_2}+\frac{0}{\mathfrak{u}_3}\big\} $	$ \big\{\frac{.5}{\mathfrak{u}_1}+\frac{.6}{\mathfrak{u}_2}+\frac{0}{\mathfrak{u}_3}\big\} $	No	-	$ \big\{\frac{0}{\mathfrak{u}_1}+\frac{0}{\mathfrak{u}_2}+\frac{0}{\mathfrak{u}_3}\big\} $	$ \big\{\frac{1}{\mathfrak{u}_1}+\frac{1}{\mathfrak{u}_2}+\frac{1}{\mathfrak{u}_3}\big\} $
$ \big\{\frac{0}{\mathfrak{u}_1}+\frac{.6}{\mathfrak{u}_2}+\frac{0}{\mathfrak{u}_3}\big\} $	$ \big\{\frac{.3}{\mathfrak{u}_1}+\frac{.6}{\mathfrak{u}_2}+\frac{0}{\mathfrak{u}_3}\big\} $	No	-	$ \big\{\frac{0}{\mathfrak{u}_1}+\frac{0}{\mathfrak{u}_2}+\frac{0}{\mathfrak{u}_3}\big\} $	$ \big\{\frac{1}{\mathfrak{u}_1}+\frac{1}{\mathfrak{u}_2}+\frac{1}{\mathfrak{u}_3}\big\} $
$ \big\{\frac{0}{\mathfrak{u}_1}+\frac{0}{\mathfrak{u}_2}+\frac{.3}{\mathfrak{u}_3}\big\} $	$ \big\{\frac{0}{\mathfrak{u}_1}+\frac{0}{\mathfrak{u}_2}+\frac{.3}{\mathfrak{u}_3}\big\} $	No	-	$ \big\{\frac{0}{\mathfrak{u}_1}+\frac{0}{\mathfrak{u}_2}+\frac{0}{\mathfrak{u}_3}\big\} $	$ \big\{\frac{1}{\mathfrak{u}_1}+\frac{1}{\mathfrak{u}_2}+\frac{1}{\mathfrak{u}_3}\big\} $
$ \big\{\frac{.4}{\mathfrak{u}_1}+\frac{.6}{\mathfrak{u}_2}+\frac{0}{\mathfrak{u}_3}\big\} $	$ \big\{\frac{.5}{\mathfrak{u}_1}+\frac{.6}{\mathfrak{u}_2}+\frac{0}{\mathfrak{u}_3}\big\} $	No	-	$ \big\{\frac{0}{\mathfrak{u}_1}+\frac{0}{\mathfrak{u}_2}+\frac{0}{\mathfrak{u}_3}\big\} $	$ \big\{\frac{1}{\mathfrak{u}_1}+\frac{1}{\mathfrak{u}_2}+\frac{1}{\mathfrak{u}_3}\big\} $
$ \big\{\frac{0}{\mathfrak{u}_1}+\frac{.6}{\mathfrak{u}_2}+\frac{.3}{\mathfrak{u}_3}\big\} $	$ \big\{\frac{.3}{\mathfrak{u}_1}+\frac{.6}{\mathfrak{u}_2}+\frac{.3}{\mathfrak{u}_3}\big\} $	No	-	$ \big\{\frac{0}{\mathfrak{u}_1}+\frac{0}{\mathfrak{u}_2}+\frac{0}{\mathfrak{u}_3}\big\} $	$ \big\{\frac{1}{\mathfrak{u}_1}+\frac{1}{\mathfrak{u}_2}+\frac{1}{\mathfrak{u}_3}\big\} $
$ \big\{\frac{.4}{\mathfrak{u}_1}+\frac{0}{\mathfrak{u}_2}+\frac{.3}{\mathfrak{u}_3}\big\} $	$ \big\{\frac{.5}{\mathfrak{u}_1}+\frac{.6}{\mathfrak{u}_2}+\frac{.3}{\mathfrak{u}_3}\big\} $	No	-	$ \big\{\frac{0}{\mathfrak{u}_1}+\frac{0}{\mathfrak{u}_2}+\frac{0}{\mathfrak{u}_3}\big\} $	$ \big\{\frac{1}{\mathfrak{u}_1}+\frac{1}{\mathfrak{u}_2}+\frac{1}{\mathfrak{u}_3}\big\} $
$ \big\{\frac{.4}{\mathfrak{u}_1}+\frac{.6}{\mathfrak{u}_2}+\frac{.3}{\mathfrak{u}_3}\big\} $	$ \big\{\frac{.5}{\mathfrak{u}_1}+\frac{1}{\mathfrak{u}_2}+\frac{.3}{\mathfrak{u}_3}\big\} $	No	-	$ \big\{\frac{0}{\mathfrak{u}_1}+\frac{0}{\mathfrak{u}_2}+\frac{0}{\mathfrak{u}_3}\big\} $	$ \big\{\frac{1}{\mathfrak{u}_1}+\frac{1}{\mathfrak{u}_2}+\frac{1}{\mathfrak{u}_3}\big\} $

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Table 2.

$ \mathcal{A}=\mathcal{P}(\mathcal{Z}_1) $	$ cl(\mathcal{A}) $	$ \tau_2-\mathfrak{K} $-FOS or not	$ \tau_2-\mathfrak{K} $-FCS	$ \underline{\mathfrak{K}}_{\mathfrak{K}}(\mathcal{A}) $	$ \overline{\mathfrak{K}}_{\mathfrak{K}}(\mathcal{A}) $
$ \big\{\frac{.4}{\mathfrak{u}_1}+\frac{0}{\mathfrak{u}_2}+\frac{0}{\mathfrak{u}_3}\big\} $	$ \big\{\frac{.5}{\mathfrak{u}_1}+\frac{.6}{\mathfrak{u}_2}+\frac{.3}{\mathfrak{u}_3}\big\} $	Yes	$ \big\{\frac{.6}{\mathfrak{u}_1}+\frac{1}{\mathfrak{u}_2}+\frac{1}{\mathfrak{u}_3}\big\} $	$ \big\{\frac{.4}{\mathfrak{u}_1}+\frac{0}{\mathfrak{u}_2}+\frac{0}{\mathfrak{u}_3}\big\} $	$ \big\{\frac{.6}{\mathfrak{u}_1}+\frac{.4}{\mathfrak{u}_2}+\frac{.7}{\mathfrak{u}_3}\big\} $
$ \big\{\frac{0}{\mathfrak{u}_1}+\frac{.6}{\mathfrak{u}_2}+\frac{0}{\mathfrak{u}_3}\big\} $	$ \big\{\frac{.5}{\mathfrak{u}_1}+\frac{.6}{\mathfrak{u}_2}+\frac{.3}{\mathfrak{u}_3}\big\} $	Yes	$ \big\{\frac{1}{\mathfrak{u}_1}+\frac{.4}{\mathfrak{u}_2}+\frac{1}{\mathfrak{u}_3}\big\} $	$ \big\{\frac{0}{\mathfrak{u}_1}+\frac{.6}{\mathfrak{u}_2}+\frac{0}{\mathfrak{u}_3}\big\} $	$ \big\{\frac{.6}{\mathfrak{u}_1}+\frac{1}{\mathfrak{u}_2}+\frac{.7}{\mathfrak{u}_3}\big\} $
$ \big\{\frac{0}{\mathfrak{u}_1}+\frac{0}{\mathfrak{u}_2}+\frac{.3}{\mathfrak{u}_3}\big\} $	$ \big\{\frac{0}{\mathfrak{u}_1}+\frac{0}{\mathfrak{u}_2}+\frac{.3}{\mathfrak{u}_3}\big\} $	No	-	$ \big\{\frac{0}{\mathfrak{u}_1}+\frac{0}{\mathfrak{u}_2}+\frac{0}{\mathfrak{u}_3}\big\} $	$ \big\{\frac{.6}{\mathfrak{u}_1}+\frac{.4}{\mathfrak{u}_2}+\frac{.7}{\mathfrak{u}_3}\big\} $
$ \big\{\frac{.4}{\mathfrak{u}_1}+\frac{.6}{\mathfrak{u}_2}+\frac{0}{\mathfrak{u}_3}\big\} $	$ \big\{\frac{.5}{\mathfrak{u}_1}+\frac{.6}{\mathfrak{u}_2}+\frac{.3}{\mathfrak{u}_3}\big\} $	Yes	$ \big\{\frac{.6}{\mathfrak{u}_1}+\frac{.4}{\mathfrak{u}_2}+\frac{1}{\mathfrak{u}_3}\big\} $	$ \big\{\frac{.4}{\mathfrak{u}_1}+\frac{.6}{\mathfrak{u}_2}+\frac{0}{\mathfrak{u}_3}\big\} $	$ \big\{\frac{.6}{\mathfrak{u}_1}+\frac{1}{\mathfrak{u}_2}+\frac{.7}{\mathfrak{u}_3}\big\} $
$ \big\{\frac{0}{\mathfrak{u}_1}+\frac{.6}{\mathfrak{u}_2}+\frac{.3}{\mathfrak{u}_3}\big\} $	$ \big\{\frac{.5}{\mathfrak{u}_1}+\frac{.6}{\mathfrak{u}_2}+\frac{.3}{\mathfrak{u}_3}\big\} $	Yes	$ \big\{\frac{1}{\mathfrak{u}_1}+\frac{.4}{\mathfrak{u}_2}+\frac{.7}{\mathfrak{u}_3}\big\} $	$ \big\{\frac{0}{\mathfrak{u}_1}+\frac{.6}{\mathfrak{u}_2}+\frac{.3}{\mathfrak{u}_3}\big\} $	$ \big\{\frac{.6}{\mathfrak{u}_1}+\frac{1}{\mathfrak{u}_2}+\frac{.7}{\mathfrak{u}_3}\big\} $
$ \big\{\frac{.4}{\mathfrak{u}_1}+\frac{0}{\mathfrak{u}_2}+\frac{.3}{\mathfrak{u}_3}\big\} $	$ \big\{\frac{.5}{\mathfrak{u}_1}+\frac{.6}{\mathfrak{u}_2}+\frac{.3}{\mathfrak{u}_3}\big\} $	Yes	$ \big\{\frac{.6}{\mathfrak{u}_1}+\frac{1}{\mathfrak{u}_2}+\frac{.7}{\mathfrak{u}_3}\big\} $	$ \big\{\frac{.4}{\mathfrak{u}_1}+\frac{0}{\mathfrak{u}_2}+\frac{.3}{\mathfrak{u}_3}\big\} $	$ \big\{\frac{.6}{\mathfrak{u}_1}+\frac{.4}{\mathfrak{u}_2}+\frac{.7}{\mathfrak{u}_3}\big\} $
$ \big\{\frac{.4}{\mathfrak{u}_1}+\frac{.6}{\mathfrak{u}_2}+\frac{.3}{\mathfrak{u}_3}\big\} $	$ \big\{\frac{.5}{\mathfrak{u}_1}+\frac{.6}{\mathfrak{u}_2}+\frac{.3}{\mathfrak{u}_3}\big\} $	Yes	$ \big\{\frac{.6}{\mathfrak{u}_1}+\frac{.4}{\mathfrak{u}_2}+\frac{.7}{\mathfrak{u}_3}\big\} $	$ \big\{\frac{.4}{\mathfrak{u}_1}+\frac{.6}{\mathfrak{u}_2}+\frac{.3}{\mathfrak{u}_3}\big\} $	$ \big\{\frac{.6}{\mathfrak{u}_1}+\frac{1}{\mathfrak{u}_2}+\frac{.7}{\mathfrak{u}_3}\big\} $

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Table 3.

$ \mathcal{A}=\mathcal{P}(\mathcal{Z}_1) $	$ cl(\mathcal{A}) $	$ \tau-\mathfrak{K} $-FOS or not	$ \tau-\mathfrak{K} $-FCS	$ \underline{\mathfrak{K}}_{\mathfrak{K}}(\mathcal{A}) $	$ \overline{\mathfrak{K}}_{\mathfrak{K}}(\mathcal{A}) $
$ \big\{\frac{.4}{\mathfrak{u}_1}+\frac{0}{\mathfrak{u}_2}+\frac{0}{\mathfrak{u}_3}\big\} $	$ \big\{\frac{.5}{\mathfrak{u}_1}+\frac{.6}{\mathfrak{u}_2}+\frac{0}{\mathfrak{u}_3}\big\} $	Yes	$ \big\{\frac{.6}{\mathfrak{u}_1}+\frac{1}{\mathfrak{u}_2}+\frac{1}{\mathfrak{u}_3}\big\} $	$ \big\{\frac{.4}{\mathfrak{u}_1}+\frac{0}{\mathfrak{u}_2}+\frac{0}{\mathfrak{u}_3}\big\} $	$ \big\{\frac{.6}{\mathfrak{u}_1}+\frac{.4}{\mathfrak{u}_2}+\frac{.7}{\mathfrak{u}_3}\big\} $
$ \big\{\frac{0}{\mathfrak{u}_1}+\frac{.6}{\mathfrak{u}_2}+\frac{0}{\mathfrak{u}_3}\big\} $	$ \big\{\frac{.3}{\mathfrak{u}_1}+\frac{.6}{\mathfrak{u}_2}+\frac{0}{\mathfrak{u}_3}\big\} $	No	-	$ \big\{\frac{0}{\mathfrak{u}_1}+\frac{0}{\mathfrak{u}_2}+\frac{0}{\mathfrak{u}_3}\big\} $	$ \big\{\frac{.6}{\mathfrak{u}_1}+\frac{1}{\mathfrak{u}_2}+\frac{.7}{\mathfrak{u}_3}\big\} $
$ \big\{\frac{0}{\mathfrak{u}_1}+\frac{0}{\mathfrak{u}_2}+\frac{.3}{\mathfrak{u}_3}\big\} $	$ \big\{\frac{0}{\mathfrak{u}_1}+\frac{0}{\mathfrak{u}_2}+\frac{.3}{\mathfrak{u}_3}\big\} $	No	-	$ \big\{\frac{0}{\mathfrak{u}_1}+\frac{0}{\mathfrak{u}_2}+\frac{0}{\mathfrak{u}_3}\big\} $	$ \big\{\frac{.6}{\mathfrak{u}_1}+\frac{.4}{\mathfrak{u}_2}+\frac{.7}{\mathfrak{u}_3}\big\} $
$ \big\{\frac{.4}{\mathfrak{u}_1}+\frac{.6}{\mathfrak{u}_2}+\frac{0}{\mathfrak{u}_3}\big\} $	$ \big\{\frac{.5}{\mathfrak{u}_1}+\frac{.6}{\mathfrak{u}_2}+\frac{0}{\mathfrak{u}_3}\big\} $	Yes	$ \big\{\frac{.6}{\mathfrak{u}_1}+\frac{.4}{\mathfrak{u}_2}+\frac{1}{\mathfrak{u}_3}\big\} $	$ \big\{\frac{.4}{\mathfrak{u}_1}+\frac{.6}{\mathfrak{u}_2}+\frac{0}{\mathfrak{u}_3}\big\} $	$ \big\{\frac{.6}{\mathfrak{u}_1}+\frac{1}{\mathfrak{u}_2}+\frac{.7}{\mathfrak{u}_3}\big\} $
$ \big\{\frac{0}{\mathfrak{u}_1}+\frac{.6}{\mathfrak{u}_2}+\frac{.3}{\mathfrak{u}_3}\big\} $	$ \big\{\frac{.3}{\mathfrak{u}_1}+\frac{.6}{\mathfrak{u}_2}+\frac{.3}{\mathfrak{u}_3}\big\} $	No	-	$ \big\{\frac{0}{\mathfrak{u}_1}+\frac{0}{\mathfrak{u}_2}+\frac{0}{\mathfrak{u}_3}\big\} $	$ \big\{\frac{.6}{\mathfrak{u}_1}+\frac{1}{\mathfrak{u}_2}+\frac{.7}{\mathfrak{u}_3}\big\} $
$ \big\{\frac{.4}{\mathfrak{u}_1}+\frac{0}{\mathfrak{u}_2}+\frac{.3}{\mathfrak{u}_3}\big\} $	$ \big\{\frac{.5}{\mathfrak{u}_1}+\frac{.6}{\mathfrak{u}_2}+\frac{.3}{\mathfrak{u}_3}\big\} $	Yes	$ \big\{\frac{.6}{\mathfrak{u}_1}+\frac{1}{\mathfrak{u}_2}+\frac{.7}{\mathfrak{u}_3}\big\} $	$ \big\{\frac{.4}{\mathfrak{u}_1}+\frac{0}{\mathfrak{u}_2}+\frac{.3}{\mathfrak{u}_3}\big\} $	$ \big\{\frac{.6}{\mathfrak{u}_1}+\frac{.4}{\mathfrak{u}_2}+\frac{.7}{\mathfrak{u}_3}\big\} $
$ \big\{\frac{.4}{\mathfrak{u}_1}+\frac{.6}{\mathfrak{u}_2}+\frac{.3}{\mathfrak{u}_3}\big\} $	$ \big\{\frac{.5}{\mathfrak{u}_1}+\frac{.6}{\mathfrak{u}_2}+\frac{.3}{\mathfrak{u}_3}\big\} $	Yes	$ \big\{\frac{.6}{\mathfrak{u}_1}+\frac{.4}{\mathfrak{u}_2}+\frac{.7}{\mathfrak{u}_3}\big\} $	$ \big\{\frac{.4}{\mathfrak{u}_1}+\frac{.6}{\mathfrak{u}_2}+\frac{.3}{\mathfrak{u}_3}\big\} $	$ \big\{\frac{.6}{\mathfrak{u}_1}+\frac{1}{\mathfrak{u}_2}+\frac{.7}{\mathfrak{u}_3}\big\} $

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Table 4.

Sr. No.	$ \tau-\mathfrak{K} $-fuzzy approximation Let $ (\hat{U}, \tau, \mathfrak{K}) $ be FTAS and $ \mathfrak{K} $ be BFR on $ \hat{U} $ and $ \mathcal{Z}_1 $ and $ \mathcal{Z}_2 $ are two fuzzy subsets of $ \hat{U}. $	Fuzzy rough approximation[4,14] Let $ \hat{U} $ be a universal set, $ \mathfrak{K} $ be BFR on $ \hat{U} $ and $ \mathcal{Z}_1 $ and $ \mathcal{Z}_2 $ are two fuzzy subsets of $ \hat{U} $.
ⅰ	$ \mathcal{Z}_1\subseteq\mathcal{Z}_2\Rightarrow \underline{\mathfrak{K}}_{\mathfrak{K}}(\mathcal{Z}_1)\subseteq\underline{\mathfrak{K}}_{\mathfrak{K}}(\mathcal{Z}_2) $ and $ \overline{\mathfrak{K}}_{\mathfrak{K}}(\mathcal{Z}_1)\subseteq\overline{\mathfrak{K}}_{\mathfrak{K}}(\mathcal{Z}_2) $	$ \mathcal{Z}_1\subseteq\mathcal{Z}_2\Rightarrow \underline{\mathfrak{K}}_{\mathfrak{K}}(\mathcal{Z}_1)\subseteq\underline{\mathfrak{K}}_{\mathfrak{K}}(\mathcal{Z}_2) $ and $ \overline{\mathfrak{K}}_{\mathfrak{K}}(\mathcal{Z}_1)\subseteq\overline{\mathfrak{K}}_{\mathfrak{K}}(\mathcal{Z}_2) $
ⅱ	$ \underline{\mathfrak{K}}_{\mathfrak{K}}(\mathcal{Z}_1)=\big( \overline{\mathfrak{K}}_{\mathfrak{K}}(\mathcal{Z}_1^c)\big)^c $	$ \underline{\mathfrak{K}}_{\mathfrak{K}}(\mathcal{Z}_1)=\big(\overline{\mathfrak{K}}_{\mathfrak{K}}(\mathcal{Z}_1^c)\big)^c $
ⅲ	$ \overline{\mathfrak{K}}_{\mathfrak{K}}(\mathcal{Z}_1)=\big(\underline{\mathfrak{K}}_{\mathfrak{K}}(\mathcal{Z}_1^c)\big)^c $	$ \overline{\mathfrak{K}}_{\mathfrak{K}}(\mathcal{Z}_1)=\big(\underline{\mathfrak{K}}_{\mathfrak{K}}(\mathcal{Z}_1^c)\big)^c $
ⅳ	$ \underline{\mathfrak{K}}_{\mathfrak{K}}(\mathcal{Z}_1\cap\mathcal{Z}_2)=\underline{\mathfrak{K}}_{\mathfrak{K}}(\mathcal{Z}_1)\cap\underline{\mathfrak{K}}_{\mathfrak{K}}(\mathcal{Z}_2) $	$ \underline{\mathfrak{K}}_{\mathfrak{K}}(\mathcal{Z}_1\cap\mathcal{Z}_2)=\underline{\mathfrak{K}}_{\mathfrak{K}}(\mathcal{Z}_1)\cap\underline{\mathfrak{K}}_{\mathfrak{K}}(\mathcal{Z}_2) $
ⅴ	$ \overline{\mathfrak{K}}_{\mathfrak{K}}(\mathcal{Z}_1\cup\mathcal{Z}_2)=\overline{\mathfrak{K}}_{\mathfrak{K}}(\mathcal{Z}_1)\cup\overline{\mathfrak{K}}_{\mathfrak{K}}(\mathcal{Z}_2) $	$ \overline{\mathfrak{K}}_{\mathfrak{K}}(\mathcal{Z}_1\cup\mathcal{Z}_2)=\overline{\mathfrak{K}}_{\mathfrak{K}}(\mathcal{Z}_1)\cup\overline{\mathfrak{K}}_{\mathfrak{K}}(\mathcal{Z}_2) $
ⅵ	$ \underline{\mathfrak{K}}_{\mathfrak{K}}(\mathcal{Z}_1)\cup\underline{\mathfrak{K}}_{\mathfrak{K}}(\mathcal{Z}_2)\subseteq\underline{\mathfrak{K}}_{\mathfrak{K}}(\mathcal{Z}_1\cup\mathcal{Z}_2) $	$ \underline{\mathfrak{K}}_{\mathfrak{K}}(\mathcal{Z}_1)\cup\underline{\mathfrak{K}}_{\mathfrak{K}}(\mathcal{Z}_2)\subseteq\underline{\mathfrak{K}}_{\mathfrak{K}}(\mathcal{Z}_1\cup\mathcal{Z}_2) $
ⅶ	$ \overline{\mathfrak{K}}_{\mathfrak{K}}(\mathcal{Z}_1\cap\mathcal{Z}_2)\subseteq\overline{\mathfrak{K}}_{\mathfrak{K}}(\mathcal{Z}_1)\cap\overline{\mathfrak{K}}_{\mathfrak{K}}(\mathcal{Z}_2) $	$ \overline{\mathfrak{K}}_{\mathfrak{K}}(\mathcal{Z}_1\cap\mathcal{Z}_2)\subseteq\overline{\mathfrak{K}}_{\mathfrak{K}}(\mathcal{Z}_1)\cap\overline{\mathfrak{K}}_{\mathfrak{K}}(\mathcal{Z}_2) $
ⅷ	$ \underline{\mathfrak{K}}_{\mathfrak{K}}(\phi)=\phi=\overline{\mathfrak{K}}_{\mathfrak{K}}(\phi) $	$ \overline{\mathfrak{K}}_{\mathfrak{K}}(\phi)=\phi $
ⅸ	$ \underline{\mathfrak{K}}_{\mathfrak{K}}(\hat{U})=\hat{U}=\overline{\mathfrak{K}}_{\mathfrak{K}}(\hat{U}) $	$ \underline{\mathfrak{K}}_{\mathfrak{K}}(\hat{U})=\hat{U} $
ⅹ	$ \underline{\mathfrak{K}}_{\mathfrak{K}}(\mathcal{Z}_1)\subseteq\mathcal{Z}_1\subseteq\overline{\mathfrak{K}}_{\mathfrak{K}}(\mathcal{Z}_1) $	$ \underline{\mathfrak{K}}_{\mathfrak{K}}(\mathcal{Z}_1)\subseteq\mathcal{Z}_1\subseteq\overline{\mathfrak{K}}_{\mathfrak{K}}(\mathcal{Z}_1) $, if $ \mathfrak{K} $ is reflexive.
xi	$ \underline{\mathfrak{K}}_{\mathfrak{K}}\bigg(\underline{\mathfrak{K}}_{\mathfrak{K}}(\mathcal{Z}_1)\bigg)=\underline{\mathfrak{K}}_{\mathfrak{K}}(\mathcal{Z}_1) $	$ \underline{\mathfrak{K}}_{\mathfrak{K}}\bigg(\underline{\mathfrak{K}}_{\mathfrak{K}}(\mathcal{Z}_1)\bigg)=\underline{\mathfrak{K}}_{\mathfrak{K}}(\mathcal{Z}_1) $, if $ \mathfrak{K} $ is reflexive and transitive.

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Table 5.

$ \mathcal{A}=\mathcal{P}(\mathcal{Z}_1) $	$ \tau-\mathfrak{K} $-FOS or not	$ \tau-\mathfrak{K} $-FCS	$ \underline{\mathfrak{K}}_{\mathfrak{K}}(\mathcal{A}_i) $	$ \overline{\mathfrak{K}}_{\mathfrak{K}}(\mathcal{A}_i) $	$ \beta_i=\frac{\underline{\mathfrak{K}}_{\mathfrak{K}}(\mathcal{A}_i)+{\overline{\mathfrak{K}}_{\mathfrak{K}}(\mathcal{A}_i)}}{2} $
$ \big\{\frac{.5}{\mathfrak{j}_2}+ \frac{0}{\mathfrak{j}_3}+\frac{0}{\mathfrak{j}_4}+\frac{0}{\mathfrak{j}_5}\big\} $	Yes	$ \big\{\frac{.5}{\mathfrak{j}_2}+ \frac{1}{\mathfrak{j}_3}+\frac{1}{\mathfrak{j}_4}+\frac{1}{\mathfrak{j}_5}\big\} $	$ \big\{\frac{.5}{\mathfrak{j}_2}+ \frac{0}{\mathfrak{j}_3}+\frac{0}{\mathfrak{j}_4}+\frac{0}{\mathfrak{j}_5}\big\} $	$ \big\{\frac{.5}{\mathfrak{j}_2}+ \frac{.5}{\mathfrak{j}_3}+\frac{.3}{\mathfrak{j}_4}+\frac{.5}{\mathfrak{j}_5}\big\} $	$ \beta_1=\big\{\frac{.5}{\mathfrak{j}_2}+ \frac{.25}{\mathfrak{j}_3}+\frac{.15}{\mathfrak{j}_4}+\frac{.25}{\mathfrak{j}_5}\big\} $
$ \big\{\frac{0}{\mathfrak{j}_2}+ \frac{.5}{\mathfrak{j}_3}+\frac{0}{\mathfrak{j}_4}+\frac{0}{\mathfrak{j}_5}\big\} $	Yes	$ \big\{\frac{1}{\mathfrak{j}_2}+ \frac{.5}{\mathfrak{j}_3}+\frac{1}{\mathfrak{j}_4}+\frac{1}{\mathfrak{j}_5}\big\} $	$ \big\{\frac{0}{\mathfrak{j}_2}+ \frac{.5}{\mathfrak{j}_3}+\frac{0}{\mathfrak{j}_4}+\frac{0}{\mathfrak{j}_5}\big\} $	$ \big\{\frac{.5}{\mathfrak{j}_2}+ \frac{.5}{\mathfrak{j}_3}+\frac{.3}{\mathfrak{j}_4}+\frac{.5}{\mathfrak{j}_5}\big\} $	$ \beta_2=\big\{\frac{.25}{\mathfrak{j}_2}+ \frac{.5}{\mathfrak{j}_3}+\frac{.15}{\mathfrak{j}_4}+\frac{.25}{\mathfrak{j}_5}\big\} $
$ \big\{\frac{0}{\mathfrak{j}_2}+ \frac{0}{\mathfrak{j}_3}+\frac{.7}{\mathfrak{j}_4}+\frac{0}{\mathfrak{j}_5}\big\} $	Yes	$ \big\{\frac{1}{\mathfrak{j}_2}+ \frac{1}{\mathfrak{j}_3}+\frac{.3}{\mathfrak{j}_4}+\frac{1}{\mathfrak{j}_5}\big\} $	$ \big\{\frac{0}{\mathfrak{j}_2}+ \frac{0}{\mathfrak{j}_3}+\frac{.7}{\mathfrak{j}_4}+\frac{0}{\mathfrak{j}_5}\big\} $	$ \big\{\frac{.5}{\mathfrak{j}_2}+ \frac{.5}{\mathfrak{j}_3}+\frac{1}{\mathfrak{j}_4}+\frac{.5}{\mathfrak{j}_5}\big\} $	$ \beta_3=\big\{\frac{.25}{\mathfrak{j}_2}+ \frac{.25}{\mathfrak{j}_3}+\frac{.85}{\mathfrak{j}_4}+\frac{.25}{\mathfrak{j}_5}\big\} $
$ \big\{\frac{0}{\mathfrak{j}_2}+ \frac{0}{\mathfrak{j}_3}+\frac{0}{\mathfrak{j}_4}+\frac{.5}{\mathfrak{j}_5}\big\} $	Yes	$ \big\{\frac{1}{\mathfrak{j}_2}+ \frac{1}{\mathfrak{j}_3}+\frac{1}{\mathfrak{j}_4}+\frac{.5}{\mathfrak{j}_5}\big\} $	$ \big\{\frac{0}{\mathfrak{j}_2}+ \frac{0}{\mathfrak{j}_3}+\frac{0}{\mathfrak{j}_4}+\frac{.5}{\mathfrak{j}_5}\big\} $	$ \big\{\frac{.5}{\mathfrak{j}_2}+ \frac{.5}{\mathfrak{j}_3}+\frac{.3}{\mathfrak{j}_4}+\frac{.5}{\mathfrak{j}_5}\big\} $	$ \beta_4=\big\{\frac{.25}{\mathfrak{j}_2}+ \frac{.25}{\mathfrak{j}_3}+\frac{.15}{\mathfrak{j}_4}+\frac{.5}{\mathfrak{j}_5}\big\} $
$ \big\{\frac{.5}{\mathfrak{j}_2}+ \frac{.5}{\mathfrak{j}_3}+\frac{0}{\mathfrak{j}_4}+\frac{0}{\mathfrak{j}_5}\big\} $	Yes	$ \big\{\frac{.5}{\mathfrak{j}_2}+ \frac{.5}{\mathfrak{j}_3}+\frac{1}{\mathfrak{j}_4}+\frac{1}{\mathfrak{j}_5}\big\} $	-	-	-
$ \big\{\frac{.5}{\mathfrak{j}_2}+ \frac{0}{\mathfrak{j}_3}+\frac{.7}{\mathfrak{j}_4}+\frac{0}{\mathfrak{j}_5}\big\} $	Yes	$ \big\{\frac{.5}{\mathfrak{j}_2}+ \frac{1}{\mathfrak{j}_3}+\frac{.3}{\mathfrak{j}_4}+\frac{1}{\mathfrak{j}_5}\big\} $	-	-	-
$ \big\{\frac{.5}{\mathfrak{j}_2}+ \frac{0}{\mathfrak{j}_3}+\frac{0}{\mathfrak{j}_4}+\frac{.5}{\mathfrak{j}_5}\big\} $	Yes	$ \big\{\frac{.5}{\mathfrak{j}_2}+ \frac{1}{\mathfrak{j}_3}+\frac{1}{\mathfrak{j}_4}+\frac{.5}{\mathfrak{j}_5}\big\} $	-	-	-
$ \big\{\frac{0}{\mathfrak{j}_2}+ \frac{.5}{\mathfrak{j}_3}+\frac{.7}{\mathfrak{j}_4}+\frac{0}{\mathfrak{j}_5}\big\} $	Yes	$ \big\{\frac{1}{\mathfrak{j}_2}+ \frac{.5}{\mathfrak{j}_3}+\frac{.3}{\mathfrak{j}_4}+\frac{1}{\mathfrak{j}_5}\big\} $	-	-	-
$ \big\{\frac{0}{\mathfrak{j}_2}+ \frac{.5}{\mathfrak{j}_3}+\frac{0}{\mathfrak{j}_4}+\frac{.5}{\mathfrak{j}_5}\big\} $	Yes	$ \big\{\frac{1}{\mathfrak{j}_2}+ \frac{.5}{\mathfrak{j}_3}+\frac{1}{\mathfrak{j}_4}+\frac{.5}{\mathfrak{j}_5}\big\} $	-	-	-
$ \big\{\frac{0}{\mathfrak{j}_2}+ \frac{0}{\mathfrak{j}_3}+\frac{.7}{\mathfrak{j}_4}+\frac{.5}{\mathfrak{j}_5}\big\} $	Yes	$ \big\{\frac{1}{\mathfrak{j}_2}+ \frac{1}{\mathfrak{j}_3}+\frac{.3}{\mathfrak{j}_4}+\frac{.5}{\mathfrak{j}_5}\big\} $	-	-	-
$ \big\{\frac{.5}{\mathfrak{j}_2}+ \frac{.5}{\mathfrak{j}_3}+\frac{.7}{\mathfrak{j}_4}+\frac{0}{\mathfrak{j}_5}\big\} $	Yes	$ \big\{\frac{.5}{\mathfrak{j}_2}+ \frac{.5}{\mathfrak{j}_3}+\frac{.3}{\mathfrak{j}_4}+\frac{1}{\mathfrak{j}_5}\big\} $	-	-	-
$ \big\{\frac{.5}{\mathfrak{j}_2}+ \frac{.5}{\mathfrak{j}_3}+\frac{0}{\mathfrak{j}_4}+\frac{.5}{\mathfrak{j}_5}\big\} $	Yes	$ \big\{\frac{.5}{\mathfrak{j}_2}+ \frac{.5}{\mathfrak{j}_3}+\frac{1}{\mathfrak{j}_4}+\frac{.5}{\mathfrak{j}_5}\big\} $	-	-	-
$ \big\{\frac{.5}{\mathfrak{j}_2}+ \frac{0}{\mathfrak{j}_3}+\frac{.7}{\mathfrak{j}_4}+\frac{.5}{\mathfrak{j}_5}\big\} $	Yes	$ \big\{\frac{.5}{\mathfrak{j}_2}+ \frac{1}{\mathfrak{j}_3}+\frac{.3}{\mathfrak{j}_4}+\frac{.5}{\mathfrak{j}_5}\big\} $	-	-	-
$ \big\{\frac{0}{\mathfrak{j}_2}+ \frac{.5}{\mathfrak{j}_3}+\frac{.7}{\mathfrak{j}_4}+\frac{.5}{\mathfrak{j}_5}\big\} $	Yes	$ \big\{\frac{1}{\mathfrak{j}_2}+ \frac{.5}{\mathfrak{j}_3}+\frac{.3}{\mathfrak{j}_4}+\frac{.5}{\mathfrak{j}_5}\big\} $	-	-	-
$ \big\{\frac{.5}{\mathfrak{j}_2}+ \frac{.5}{\mathfrak{j}_3}+\frac{.7}{\mathfrak{j}_4}+\frac{.5}{\mathfrak{j}_5}\big\} $	Yes	$ \big\{\frac{.5}{\mathfrak{j}_2}+ \frac{.5}{\mathfrak{j}_3}+\frac{.3}{\mathfrak{j}_4}+\frac{.5}{\mathfrak{j}_5}\big\} $	-	-	-

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