The theory of rough set represents a non-statistical methodology for analyzing ambiguity and imprecise information. It can be characterized by two crisp sets, named the upper and lower approximations that are used to determine the boundary region and accurate measure of any subset. This article endeavors to achieve the best approximation and the highest accuracy degree by using the minimal structure approximation space

Topological structures and their generalizations are of crucial importance in data analysis, which have manifested in different fields, for example, in physics [

The main contributions of this work are constructing ideal minimal structure approximation space

Herein, some vital concepts and results are introduced, which are helpful in the sequel.

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The sets of all

This section aims to introduce the ideal minimal structure approximation space

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To study the prime properties of

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The succeeding proposition is realized depending on Proposition 1.

The next remark is devoted to clarifying the differences between the existing approximations and the preceding one of Propositions 2 & 3 in [

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A method of Definition 4 | Current method of Definition 6 | |||||
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0 | 0 | |||||

0 | 0.5 | |||||

1 | 1 | |||||

1 | 1 | |||||

1 | 1 | |||||

0.33 | 0.5 | |||||

0.5 | 0.5 | |||||

0.67 | 0.67 | |||||

0.33 | 0.67 | |||||

1 | 1 | |||||

1 | 1 | |||||

1 | 1 | |||||

0.5 | 0.67 | |||||

0.75 | 0.75 | |||||

1 | 1 |

A method of Definition 4 | Current method of Definition 6 | |||||
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{a} | {a} | {a} | {a} | {a} | ||

{b} | {a, b} | {b} | {a} | {a} | {a} | |

{c} | ||||||

{d} | ||||||

{a, b} | ||||||

{a, c} | {a, b} | {a} | {b} | {a} | {a} | |

{a, d} | {a} | {a} | {a} | {a} | ||

{b, c} | {a} | {a} | {a} | {a} | ||

{b, d} | {a, b} | {b} | {a} | {a} | {a} | |

{c, d} | ||||||

{a, b, c} | ||||||

{a, b, d} | ||||||

{a, c, d} | {a, b} | {a} | {b} | {a} | {a} | |

{b, c, d} | {a} | {a} | {a} | {a} | ||

The next result exhibits the connections among the lower, and upper approximations and the degree of accuracy that were offered in both Definitions 4, and 6.

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This section aims to introduce and investigate some sorts of near open (resp. closed) sets via the viewpoint of an

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According to Definition 9, the proof of the proposition 5 is obvious.

The next example shows that the converse of Proposition 5 is incorrect.

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An example in the area of chemistry is provided by utilizing the actual approximation in Definition 6 to clarify the notions practically.

0.23 | 254.2 | 2.126 | −0.02 | 82.2 | |

−0.48 | 303.6 | 2.994 | −1.24 | 112.3 | |

−0.61 | 287.9 | 2.994 | −1.08 | 103.7 | |

0.45 | 282.9 | 2.933 | −0.11 | 99.1 | |

−0.11 | 335.0 | 3.458 | −0.19 | 127.5 |

Presently, it shall be investigated five relations on

The intersection of all right neighborhoods of all elements

If

0 | 0 | 0 | 0.33 | ||

0 | 0.5 | 0.5 | 0.75 | ||

0 | 0 | 0.5 | 0.5 | ||

0.33 | 0.33 | 0.67 | 0.67 | ||

0.33 | 1 | 0.25 | 0.33 | ||

0 | 0.33 | 0.6 | 0.75 | ||

0 | 0 | 0.25 | 0.75 | ||

0.67 | 0.67 | 0.67 | 0.67 | ||

0.25 | 0.5 | 0.6 | 0.6 | ||

0 | 0 | 0.4 | 0.5 | ||

0.25 | 0.5 | 0.5 | 1 | ||

0.67 | 0.67 | 0.5 | 0.5 | ||

0.33 | 0.33 | 0.8 | 0.8 | ||

0.33 | 0.5 | 0 | 0.75 | ||

0.4 | 0.5 | 0.8 | 0.8 |

The ideal minimal accuracy measure

This section provides an algorithm and a framework for decision-making problems. The suggested algorithm is checked with fictitious data and compared to existing methods. This technique represents a simple tool that can be used in MATLAB.

The following figure (

The novel rough approximation space

One of the challenges in daily problems, as in the medical diagnosis, is making an accurate decision. Therefore, the applied example in biochemistry offers a clear vision that the expansion using the ideal gives better results. Thus, by the

In the forthcoming, the

We appreciate the reviewers for their invaluable time in reviewing our paper and providing thoughtful and valuable comments. It was their insightful suggestions that led to sensible improvements in the current version.

The authors received no specific funding for this study.

The authors declare that they have no conflicts of interest to report regarding the present study.

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