﻿ 多粒度近似空间中基于“逻辑或”算子的双量化粗糙集模型 Double Quantitative Rough Set Model Based on Logical Disjunct Operation in Multi-Granulation Approximate Space

Operations Research and Fuzziology
Vol.07 No.04(2017), Article ID:22781,11 pages
10.12677/ORF.2017.74016

Double Quantitative Rough Set Model Based on Logical Disjunct Operation in Multi-Granulation Approximate Space

Huafeng Chen1, Yuling Shen1, Xianping Qu2

1Department of Foundation, Chongqing Telecommunication Polytechnic College, Chongqing

2College of Computer Science and Engineering, Chongqing University of Technology, Chongqing

Received: Nov. 4th, 2017; accepted: Nov. 16th, 2017; published: Nov. 24th, 2017

ABSTRACT

Both the variable precision rough set and graded rough set are the generalized rough set models based on indiscernibility relation in single granulation approximate space. In the viewpoint of information quantitative, the variable precision rough set describes relative quantitative information, while graded rough set is used to represent the absolute quantitative information in approximate space. The multi-granulation approximate space as a generalized approximate space is a natural expansion of classical approximation space. In order to study the rough set model that include characteristics of variable precision rough set and graded rough set. We combine them into a double-quantitative multi-granulation rough set model based on logical disjunct operation. Furthermore, the rough set regions and some basic mathematical properties of the proposed model are discussed in detail. This research provides a novel approach to knowledge discovery in multi-granulation approximate space based on rough set theory.

Keywords:Multi-Granulation Approximate Space, Logical Disjunct Operation, Variable Precision Rough Set, Graded Rough Set, Double Quantitative

1重庆电讯职业学院，基础部，重庆

2重庆理工大学，计算机科学与工程学院，重庆

Copyright © 2017 by authors and Hans Publishers Inc.

1. 引言

20世纪80年代波兰数学家Pawlak在经典集合论的基础上，提出了粗糙集理论 [1] [2] ，对知识的自动获取、机器学习以及模式识别等多个科学研究领域的发展都起到了积极的推动作用。粗糙集的理论基础是基于一族等价关系所构成的不可分辨关系对论域的划分，然后利用集合间的包含关系定义了一对近似算子，再利用近似算子根据已有的知识来近似地逼近未知的概念。该理论将人的智能体现在对事物、行为、感知等的分类能力上，而不确定性正好可以归属到这对近似集所刻画的边界域里。

2. 预备知识

2.1. 变精度粗糙集模型

$S$ 为一个信息系统，对任意的 $X\subseteq U$$A\subseteq AT$$c\left({\left[x\right]}_{A},X\right)=1-|{\left[x\right]}_{A}\cap X|/|{\left[x\right]}_{A}|$ 称为等价类 ${\left[x\right]}_{A}$ 关于集合 $X$ 的错误分分类率，其中 $|\text{ }·\text{ }|$ 表示集合的势。设 $\beta \in \left[0,0.5\right)$ 称为可调错误分类水平， $1-\beta$ 称为精度。集合

$\overline{{A}_{\beta }}X=\left\{x\in U:\text{\hspace{0.17em}}c\left({\left[x\right]}_{A},X\right)<1-\beta \right\}$

$\underset{_}{{A}_{\beta }}X=\left\{x\in U:\text{\hspace{0.17em}}c\left({\left[x\right]}_{A},X\right)\le \beta \right\}$

2.2. 程度粗糙集模型

$S$ 为一个信息系统，对任意的 $X\subseteq U$$A\subseteq AT$$k$ (非负常数)为自然数，集合

$\overline{{A}_{k}}X=\left\{x\in U:|{\left[x\right]}_{A}\cap X|>k\right\}$

$\underset{_}{{A}_{k}}X=\left\{x\in U:|{\left[x\right]}_{A}|-|{\left[x\right]}_{A}\cap X|\le k\right\}$

2.3. 多粒度粗糙集模型

$S$ 为一个信息系统， ${A}_{1},{A}_{2},\cdots ,{A}_{m}\subseteq AT$ ，任意的 $X\subseteq U$$X$ 的乐观多粒度下近似算子 $\underset{_}{\sum _{i=1}^{m}{A}_{i}^{O}}\left(X\right)$ 与上近似算子 $\overline{\sum _{i=1}^{m}{A}_{i}^{O}}\left(X\right)$ 分别定义为：

$\underset{_}{\sum _{i=1}^{m}{A}_{i}^{O}}\left(X\right)=\left\{x\in U:{\left[x\right]}_{{A}_{1}}\subseteq X\vee \cdots \vee {\left[x\right]}_{{A}_{m}}\subseteq X\right\}$

$\overline{\sum _{i=1}^{m}{A}_{i}^{O}}\left(X\right)=\sim \underset{_}{\sum _{i=1}^{m}{A}_{i}^{O}}\left(\sim X\right)$

$\underset{_}{\sum _{i=1}^{m}{A}_{i}^{P}}\left(X\right)=\left\{x\in U:{\left[x\right]}_{{A}_{1}}\subseteq X\wedge \cdots \wedge {\left[x\right]}_{{A}_{m}}\subseteq X\right\}$

$\overline{\sum _{i=1}^{m}{A}_{i}^{P}}\left(X\right)=\sim \underset{_}{\sum _{i=1}^{m}{A}_{i}^{P}}\left(\sim X\right)$

3. 多粒度近似空间中的“逻辑或”双量化粗糙集模型

3.1. “逻辑或”双量化乐观多粒度粗糙集模型

$\underset{_}{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee k}^{O}}\left(X\right)=\left\{x\in U:\underset{j=1}{\overset{m}{\vee }}\left(c\left({\left[x\right]}_{{A}_{j}},X\right)\le \beta \right)\vee \underset{l=1}{\overset{m}{\vee }}\left(|{\left[x\right]}_{{A}_{l}}|-|{\left[x\right]}_{{A}_{l}}\cap X|\le k\right)\right\}$

$\overline{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee k}^{O}}\left(X\right)=~\underset{_}{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee k}^{O}}\left(~X\right)$

1) $Pos{\left(X\right)}_{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee k}^{O}}=\underset{_}{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee k}^{O}}\left(X\right)\cap \overline{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee k}^{O}}\left(X\right)$

2) $Neg{\left(X\right)}_{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee k}^{O}}=\sim \left(\underset{_}{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee k}^{O}}\left(X\right)\cup \overline{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee k}^{O}}\left(X\right)\right)$

3) $Ubn{\left(X\right)}_{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee k}^{O}}=\overline{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee k}^{O}}\left(X\right)-\underset{_}{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee k}^{O}}\left(X\right)$

4) $Lbn{\left(X\right)}_{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee k}^{O}}=\underset{_}{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee k}^{O}}\left(X\right)-\overline{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee k}^{O}}\left(X\right)$

5) $Bn{\left(X\right)}_{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee k}^{O}}=\overline{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee k}^{O}}\left(X\right)\text{\hspace{0.17em}}\Delta \text{\hspace{0.17em}}\underset{_}{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee k}^{O}}\left(X\right)$

$\overline{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee k}^{O}}\left(X\right)=\left\{x\in U:\underset{j=1}{\overset{m}{\wedge }}\left(c\left({\left[x\right]}_{{A}_{j}},X\right)<1-\beta \right)\wedge \underset{l=1}{\overset{m}{\wedge }}\left(|{\left[x\right]}_{{A}_{l}}\cap X|>k\right)\right\}$

$\begin{array}{l}\overline{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee k}^{O}}\left(X\right)=~\underset{_}{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee k}^{O}}\left(~X\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\sim \left\{x\in U:\underset{l=1}{\overset{m}{\vee }}\left(|{\left[x\right]}_{{A}_{l}}|-|{\left[x\right]}_{{A}_{l}}\cap \sim X|\le k\right)\vee \underset{j=1}{\overset{m}{\vee }}\left(c\left({\left[x\right]}_{{A}_{j}},\sim X\right)\right)\le \beta \right\}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\left\{x\in U:\underset{j=1}{\overset{m}{\wedge }}\left(c\left({\left[x\right]}_{{A}_{j}},\sim X\right)>\beta \right)\wedge \underset{l=1}{\overset{m}{\wedge }}\left(|{\left[x\right]}_{{A}_{l}}|-|{\left[x\right]}_{{A}_{l}}\cap \sim X|>k\right)\right\}\end{array}$

$\begin{array}{l}=\left\{x\in U:\underset{j=1}{\overset{m}{\wedge }}\left(1-\frac{|{\left[x\right]}_{{A}_{j}}\cap \sim X|}{|{\left[x\right]}_{{A}_{j}}|}>\beta \right)\wedge \underset{l=1}{\overset{m}{\wedge }}\left(|{\left[x\right]}_{{A}_{l}}|-|{\left[x\right]}_{{A}_{l}}\cap \sim X|>k\right)\right\}\\ =\left\{x\in U:\underset{j=1}{\overset{m}{\wedge }}\left(\frac{|{\left[x\right]}_{{A}_{j}}\cap X|}{|{\left[x\right]}_{{A}_{j}}|}>\beta \right)\wedge \left(\underset{l=1}{\overset{m}{\wedge }}|{\left[x\right]}_{{A}_{l}}\cap X|>k\right)\right\}\\ =\left\{x\in U:\underset{j=1}{\overset{m}{\wedge }}\left(c\left({\left[x\right]}_{{A}_{j}},X\right)<1-\beta \right)\wedge \underset{l=1}{\overset{m}{\wedge }}\left(|{\left[x\right]}_{{A}_{l}}\cap X|>k\right)\right\}\end{array}$

1) $\underset{_}{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee k}^{O}}\left(U\right)=U$

2) $\overline{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee k}^{O}}\left(\varphi \right)=\varphi$

3) $\underset{_}{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee k}^{O}}\left(X\cap Y\right)\subseteq \underset{_}{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee k}^{O}}\left(X\right)\cap \underset{_}{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee k}^{O}}\left(Y\right)$

4) $\underset{_}{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee k}^{O}}\left(X\cup Y\right)\supseteq \underset{_}{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee k}^{O}}\left(X\right)\cup \underset{_}{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee k}^{O}}\left(Y\right)$

5) $\overline{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee k}^{O}}\left(X\cap Y\right)\subseteq \overline{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee k}^{O}}\left(X\right)\cap \underset{_}{\overline{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee k}^{O}}}\left(Y\right)$

6) $\overline{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee k}^{O}}\left(X\cup Y\right)\supseteq \overline{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee k}^{O}}\left(X\right)\cup \underset{_}{\overline{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee k}^{O}}}\left(Y\right)$

7) 若 ${k}_{1}\le {k}_{2}$ ，则 $\underset{_}{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee {k}_{1}}^{O}}\left(X\right)\subseteq \underset{_}{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee {k}_{2}}^{O}}\left(X\right)$$\overline{{\sum _{i=1}^{m}{A}_{i}}_{\beta \wedge {k}_{2}}^{O}}\left(X\right)\subseteq \overline{{\sum _{i=1}^{m}{A}_{i}}_{\beta \wedge {k}_{1}}^{O}}\left(X\right)$

8) 若 ${\beta }_{1}\le {\beta }_{2}\in \left[0,0.5\right)$ ，则 $\underset{_}{{\sum _{i=1}^{m}{A}_{i}}_{{\beta }_{1}\vee k}^{O}}\left(X\right)\subseteq \underset{_}{{\sum _{i=1}^{m}{A}_{i}}_{{\beta }_{2}\vee k}^{O}}\left(X\right)$$\overline{{\sum _{i=1}^{m}{A}_{i}}_{{\beta }_{2}\vee k}^{O}}\left(X\right)\subseteq \overline{{\sum _{i=1}^{m}{A}_{i}}_{{\beta }_{1}\vee k}^{O}}\left(X\right)$

3) 对任意的 $x\in \underset{_}{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee k}^{O}}\left(X\cap Y\right)$ ，由定义3.1.1可知存在粒度 ${A}_{j},{A}_{l}$ 使得 $c\left({\left[x\right]}_{{A}_{j}},X\cap Y\right)\le \beta$ 或者 $|{\left[x\right]}_{{A}_{l}}|-|{\left[x\right]}_{{A}_{l}}\cap \left(X\cap Y\right)|\le k$ ，其中 ${A}_{j}$${A}_{l}$ 无关。又因为对任意的 $X,Y\subseteq U$ 有， $c\left({\left[x\right]}_{{A}_{j}},X\right)\le c\left({\left[x\right]}_{{A}_{j}},X\cap Y\right)\le \beta$ 或者 $|{\left[x\right]}_{{A}_{l}}|-|{\left[x\right]}_{{A}_{l}}\cap X|\le |{\left[x\right]}_{{A}_{l}}|-|{\left[x\right]}_{{A}_{l}}\cap \left(X\cap Y\right)|\le k$ ，也就是有 $x\in \underset{_}{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee k}^{O}}\left(X\right)$ ，同理可得 $x\in \underset{_}{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee k}^{O}}\left(Y\right)$ 。所以 $x\in \underset{_}{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee k}^{O}}\left(X\right)\cap \underset{_}{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee k}^{O}}\left(Y\right)$ ，即 $\underset{_}{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee k}^{O}}\left(X\cap Y\right)\subseteq \underset{_}{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee k}^{O}}\left(X\right)\cap \underset{_}{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee k}^{O}}\left(Y\right)$

4) 任取 $x\in \left(\underset{_}{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee k}^{O}}\left(X\right)\cup \underset{_}{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee k}^{O}}\left(Y\right)\right)$ ，若 $x\in \underset{_}{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee k}^{O}}\left(X\right)$$c\left({\left[x\right]}_{{A}_{j}},X\right)\le \beta$ 或者 $|{\left[x\right]}_{{A}_{l}}|-|{\left[x\right]}_{{A}_{l}}\cap X|\le k$ ，任 $Y\subseteq U$$c\left({\left[x\right]}_{{A}_{j}},X\cup Y\right)\le c\left({\left[x\right]}_{{A}_{j}},X\right)\le \beta$ ，或者 $|{\left[x\right]}_{{A}_{l}}|-|{\left[x\right]}_{{A}_{l}}\cap \left(X\cup Y\right)|\le |{\left[x\right]}_{{A}_{l}}|-|{\left[x\right]}_{{A}_{l}}\cap X|\le k$ ，则有 $x\in \underset{_}{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee k}^{O}}\left(X\cup Y\right)$$x\in \underset{_}{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee k}^{O}}\left(Y\right)$$x\in \underset{_}{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee k}^{O}}\left(X\cup Y\right)$ ，则 $\underset{_}{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee k}^{O}}\left(X\cup Y\right)\supseteq \underset{_}{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee k}^{O}}\left(X\right)\cup \underset{_}{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee k}^{O}}\left(Y\right)$

5) 任取 $x\in \overline{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee k}^{O}}\left(X\cap Y\right)$ ，由定理3.1.1知对任意的粒度 ${A}_{j},{A}_{l}$ ，有 $c\left({\left[x\right]}_{{A}_{j}},X\cap Y\right)<1-\beta$$|{\left[x\right]}_{{A}_{l}}\cap \left(X\cap Y\right)|>k$ ，又 $c\left({\left[x\right]}_{{A}_{j}},X\right)\le c\left({\left[x\right]}_{{A}_{j}},X\cap Y\right)<1-\beta$ ，同时 $|{\left[x\right]}_{{A}_{l}}\cap X|\ge |{\left[x\right]}_{{A}_{l}}\cap \left(X\cap Y\right)|>k$ ，所以有 $x\in \overline{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee k}^{O}}\left(X\right)$ ，同理可得 $x\in \overline{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee k}^{O}}\left(Y\right)$ 也同时成立，因此(5)得证。

6) 由定义3.1.1可知，上下近似算子之间存在对偶性，因此由(5)的证明可知(6)也成立。

7) 不妨任取 $x\in \underset{_}{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee {k}_{1}}^{O}}\left(X\right)$ ，由定义知存在粒度 ${A}_{j}$${A}_{l}$ ，使得 $c\left({\left[x\right]}_{{A}_{j}},X\right)\le \beta$ 或者 $|{\left[x\right]}_{{A}_{l}}|-|{\left[x\right]}_{{A}_{l}}\cap X|\le {k}_{1}$ ，又因 ${k}_{1}\le {k}_{2}$ ，故 $|{\left[x\right]}_{{A}_{l}}|-|{\left[x\right]}_{{A}_{l}}\cap X|\le {k}_{2}$ ，则 $x\in \underset{_}{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee {k}_{2}}^{O}}\left(X\right)$ ，也即是 $\underset{_}{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee {k}_{1}}^{O}}\left(X\right)\subseteq \underset{_}{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee {k}_{2}}^{O}}\left(X\right)$ 。那么可以得 $\underset{_}{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee {k}_{1}}^{O}}\left(\sim X\right)\subseteq \underset{_}{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee {k}_{2}}^{O}}\left(\sim X\right)$ ，那么 $\sim \underset{_}{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee {k}_{1}}^{O}}\left(\sim X\right)\supseteq \sim \underset{_}{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee {k}_{2}}^{O}}\left(\sim X\right)$ ，由定义可知 $\overline{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee {k}_{2}}^{O}}\left(X\right)\subseteq \overline{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee {k}_{1}}^{O}}\left(X\right)$

8) 对任意的 $x\in \underset{_}{{\sum _{i=1}^{m}{A}_{i}}_{{\beta }_{1}\vee k}^{O}}\left(X\right)$ ，有 $c\left({\left[x\right]}_{{A}_{j}},X\right)\le {\beta }_{1}$ 或者 $|{\left[x\right]}_{{A}_{l}}|-|{\left[x\right]}_{{A}_{l}}\cap X|\le k$ ，而 ${\beta }_{1}\le {\beta }_{2}$ ，则 $c\left({\left[x\right]}_{{A}_{j}},X\right)\le {\beta }_{1}\le {\beta }_{2}$ ，所以有 $x\in \underset{_}{{\sum _{i=1}^{m}{A}_{i}}_{{\beta }_{2}\vee k}^{O}}\left(X\right)$ ，即 $\underset{_}{{\sum _{i=1}^{m}{A}_{i}}_{{\beta }_{1}\vee k}^{O}}\left(X\right)\subseteq \underset{_}{{\sum _{i=1}^{m}{A}_{i}}_{{\beta }_{2}\vee k}^{O}}\left(X\right)$ 。与(7)类似，利用上下近似算子间的对偶性可直接得到 $\overline{{\sum _{i=1}^{m}{A}_{i}}_{{\beta }_{2}\vee k}^{O}}\left(X\right)\subseteq \overline{{\sum _{i=1}^{m}{A}_{i}}_{{\beta }_{1}\vee k}^{O}}\left(X\right)$

3.2. “逻辑或”双量化悲观多粒度粗糙集模型

$\underset{_}{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee k}^{P}}\left(X\right)=\left\{x\in U:\underset{j=1}{\overset{m}{\wedge }}\left(c\left({\left[x\right]}_{{A}_{j}},X\right)\le \beta \right)\vee \underset{l=1}{\overset{m}{\wedge }}\left(|{\left[x\right]}_{{A}_{l}}|-|{\left[x\right]}_{{A}_{l}}\cap X|\le k\right)\right\}$

$\overline{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee k}^{P}}\left(X\right)=~\underset{_}{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee k}^{P}}\left(~X\right)$

$\overline{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee k}^{P}}\left(X\right)=\left\{x\in U:\underset{j=1}{\overset{m}{\vee }}\left(c\left({\left[x\right]}_{{A}_{j}},X\right)<1-\beta \right)\wedge \underset{l=1}{\overset{m}{\vee }}\left(|{\left[x\right]}_{{A}_{l}}\cap X|>k\right)\right\}$

1) $\underset{_}{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee k}^{P}}\left(U\right)=U$

2) $\overline{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee k}^{P}}\left(\varphi \right)=\varphi$

3) $\underset{_}{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee k}^{P}}\left(X\cap Y\right)\subseteq \underset{_}{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee k}^{P}}\left(X\right)\cap \underset{_}{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee k}^{P}}\left(Y\right)$

4) $\underset{_}{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee k}^{P}}\left(X\cup Y\right)\supseteq \underset{_}{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee k}^{P}}\left(X\right)\cup \underset{_}{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee k}^{P}}\left(Y\right)$

5) $\overline{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee k}^{P}}\left(X\cap Y\right)\subseteq \overline{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee k}^{P}}\left(X\right)\cap \underset{_}{\overline{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee k}^{P}}}\left(Y\right)$

6) $\overline{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee k}^{P}}\left(X\cup Y\right)\supseteq \overline{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee k}^{P}}\left(X\right)\cup \underset{_}{\overline{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee k}^{P}}}\left(Y\right)$

7) 若 ${k}_{1}\le {k}_{2}$ ，则 $\underset{_}{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee {k}_{1}}^{P}}\left(X\right)\subseteq \underset{_}{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee {k}_{2}}^{P}}\left(X\right)$$\overline{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee {k}_{2}}^{P}}\left(X\right)\subseteq \overline{{\sum _{i=1}^{m}{A}_{i}}_{\beta \vee {k}_{1}}^{P}}\left(X\right)$

8) 若 ${\beta }_{1}\le {\beta }_{2}$ ，则 $\underset{_}{{\sum _{i=1}^{m}{A}_{i}}_{{\beta }_{1}\vee k}^{P}}\left(X\right)\subseteq \underset{_}{{\sum _{i=1}^{m}{A}_{i}}_{{\beta }_{2}\vee k}^{P}}\left(X\right)$$\overline{{\sum _{i=1}^{m}{A}_{i}}_{{\beta }_{2}\vee k}^{P}}\left(X\right)\subseteq \overline{{\sum _{i=1}^{m}{A}_{i}}_{{\beta }_{1}\vee k}^{P}}\left(X\right)$

4. 案例分析

${A}_{1}=\left\{{a}_{1},{a}_{2}\right\}$${A}_{2}=\left\{{a}_{2},{a}_{3}\right\}$${A}_{3}=\left\{{a}_{3},{a}_{4}\right\}$ 表示三个不同粒度，则该信息系统关于这三个不同的粒度得到划分如下：

$U/{A}_{1}=\left\{\left\{{x}_{1},{x}_{6},{x}_{7},{x}_{10}\right\},\left\{{x}_{2},{x}_{5}\right\},\left\{{x}_{3},{x}_{8}\right\},\left\{{x}_{4},{x}_{9}\right\}\right\}$

$U/{A}_{2}=\left\{\left\{{x}_{1},{x}_{9},{x}_{10}\right\},\left\{{x}_{2},{x}_{5}\right\},\left\{{x}_{3},{x}_{8}\right\},\left\{{x}_{4},{x}_{6},{x}_{7}\right\}\right\}$

$U/{A}_{3}=\left\{\left\{{x}_{1},{x}_{2},{x}_{10}\right\},\left\{{x}_{3},{x}_{8}\right\},\left\{{x}_{4}\right\},\left\{{x}_{5},{x}_{9}\right\},\left\{{x}_{6},{x}_{7}\right\}\right\}$

$X=\left\{{x}_{1},{x}_{4},{x}_{5},{x}_{6},{x}_{7}\right\}$ ，则集合 $X$ 在粒度 ${A}_{i}\left(i=1,2,3\right)$ 下与 ${\left[{x}_{j}\right]}_{{A}_{i}}$ 的关系如表2所示。

$\overline{{\sum _{i=1}^{3}{A}_{i}}_{0.35\vee 1}^{O}}\left(X\right)=\varphi$

Table 1. A real estate investment information system

Table 2. The structure of information systems in granularity A i ( i = 1 , 2 , 3 )

$\underset{_}{{\sum _{i=1}^{3}{A}_{i}}_{0.35\vee 1}^{O}}\left(X\right)=\left\{{x}_{1},{x}_{2},{x}_{4},{x}_{5},{x}_{6},{x}_{7},{x}_{9},{x}_{10}\right\}$

$\overline{{\sum _{i=1}^{3}{A}_{i}}_{0.35\vee 1}^{P}}\left(X\right)=\left\{{x}_{1},{x}_{4},{x}_{6},{x}_{7},{x}_{10}\right\}$

$\underset{_}{{\sum _{i=1}^{3}{A}_{i}}_{0.35\vee 1}^{P}}\left(X\right)=\left\{{x}_{4},{x}_{5},{x}_{6},{x}_{7}\right\}$

5. 结束语

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