﻿ 一种参照模糊集的云模型集合论方法研究
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 智能系统学报  2020, Vol. 15 Issue (3): 507-513  DOI: 10.11992/tis.201810030 0

引用本文

WANG Hongli. A method of cloud model set theory referring to fuzzy sets[J]. CAAI Transactions on Intelligent Systems, 2020, 15(3): 507-513. DOI: 10.11992/tis.201810030.

文章历史

A method of cloud model set theory referring to fuzzy sets
WANG Hongli
School of Economic and Trade, Fujian Jiangxia University, Fuzhou 350108, China
Abstract: Fuzzy sets have been extensively and deeply studied and applied. The set theory method of building cloud models with reference to fuzzy sets can well extend the application field of cloud models. In this paper, a cloud model theory based on fuzzy sets is proposed. First, the cloud model and its constituent elements are described. Afterward, the I and P operations of the cloud set elements are proposed. Then, the basic operation method of the cloud set is given. Finally, the cut set and the theorem of decomposition of the cloud set are studied. The research has significance as a good reference for extension of cloud models in the set theory.
Key words: set theory    cloud model    cloud set    fuzzy set    cut set    cloud fraction theorem    I operation    P operation

1 云集合及其组成元素 1.1 云集合

1.2 云集合与模糊集的关系

 $\mathop A\limits_\wp = \Bigg\{ \mathop x\limits^\wp \left| {{u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )} \right. = {{\rm{e}}^{ - \frac{{{{(\mathop x\limits^\wp - {\rm{E}}x)}^2}}}{{2{\rm{E}}n{{_j'}^2}}}}}{\rm{, }}{\rm{E}}n_j' = {\rm{Nor}}({\rm{E}}n,He)\Bigg\}$

 $\mathop A\limits_{\sim} = \Bigg\{ \left. {\mathop x\limits^{\sim} } \right|{u_{\mathop A\limits_{\sim} }}(\mathop x\limits^{\sim} ) = {{\rm{e}}^{ - \frac{{{{(\mathop x\limits^{\sim} - {\rm{E}}x)}^2}}}{{2{{({\rm{Cer}}({\rm{E}}n_j'))}^2}}}}}\Bigg\}$

1.3 云集合的组成元素

1)首先生成以 ${\rm{E}}{n_j}$ 为期望、 $H{e_j}$ 为标准差的随机数 ${\rm{E}}n_j'$

2)令 ${u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp ) = {{\rm{e}}^{ - \frac{{{{(\mathop x\limits^\wp - {\rm{E}}x)}^2}}}{{2{\rm{E}}n{{_j'}^2}}}}}$ 为云集合元素 $\mathop x\limits^\wp$ 的随机隶属度。

 Download: 图 1 具有随机隶属度的云集合元素 Fig. 1 Element of cloud set with random subjection degree
2 云集合的基础运算方法 2.1 云集合元素的I运算和P运算 2.1.1 云集合元素 $\mathop x\limits^\wp$ $I$ 运算

$I$ 运算是求随机隶属度取值区间的运算，即求云集合元素 $\mathop x\limits^\wp$ 的随机隶属度 ${u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )$ 的取值区间，记为 $I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp ))$ 。对于正态云集合，集合元素 $\mathop x\limits^\wp$ $I$ 运算的方法为

 $I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )) = \left[ {{{\rm{e}}^{ - \frac{{{{(\mathop x\limits^\wp - {\rm{E}}{x_A})}^2}}}{{2{{({\rm{E}}{n_A}{\rm{ - }}3H{{\rm{e}}_A})}^2}}}}},\;{{\rm{e}}^{ - \frac{{{{(\mathop x\limits^\wp - {\rm{E}}{x_A})}^2}}}{{2{{({\rm{E}}{n_A} + 3H{{\rm{e}}_A})}^2}}}}}} \right],\quad \forall \mathop x\limits^\wp \in \mathop A\limits_\wp$ (1)

 $I({\rm{E}}n_j') = [{\rm{E}}n - 3He,{\rm{E}}n + 3He]$
2.1.2 云集合元素 $\mathop x\limits^\wp$ $P$ 运算

$P$ 运算是计算一个取值区间大于另一个取值区间的可能性的运算，记作 $P({I_1} \geqslant {I_2})$ $\mathop A\limits_\wp$ $\mathop B\limits_\wp$ 是论域 $U$ 上的云子集， $P(I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )) \geqslant I({u_{\mathop B\limits_\wp }}(\mathop x\limits^\wp )))$ 表示区间 $I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp ))$ 不小于区间 $I({u_{\mathop B\limits_\wp }}(\mathop x\limits^\wp ))$ 的可能性，计算方法为：设 $I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp ))$ $I({u_{\mathop B\limits_\wp }}(\mathop x\limits^\wp ))$ 使用区间表示为 $[{a^l},{a^r}]$ $[{b^l},{b^r}]$ ，则

 $\begin{array}{l} P(I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )) \geqslant I({u_{\mathop B\limits_\wp }}(\mathop x\limits^\wp ))) = P([{a^l},{b^r}] \geqslant [{a^l},{b^r}]) = \\ \left\{ {\begin{array}{*{20}{l}} 1,\quad{{a^l} \geqslant {b^r}} \\ {\dfrac{{{a^r} - {b^r}}}{{{a^r} - {b^l}}} + \dfrac{{{b^r} - {a^l}}}{{2({a^r} - {b^l})}}},\quad{{b^l} < {a^l} < {b^r} < {a^r}} \\ {\dfrac{1}{2}},\quad{{a^l} = {b^l},{a^r} = {b^r}} \\ {\dfrac{{{a^r} - {b^l}}}{{2({b^r} - {a^l})}}},\quad{{a^l} < {b^l} < {a^r} < {b^r}} \\ {\dfrac{{{a^r} - {b^r}}}{{{a^r} - {a^l}}} + \dfrac{{{b^r} - {b^l}}}{{2({a^r} - {a^l})}}},\quad{{a^l} < {b^l} < {b^r} < {a^r}} \\ {\dfrac{{{a^r} - {a^l}}}{{2({b^r} - {b^l})}}},\quad{{b^l} < {a^l} < {a^r} < {b^r}} \\ 0,\quad{{b^l} \geqslant {a^r}} \end{array}} \right. \\ \end{array}$
2.2 云集合的基础运算

$\mathop A\limits_\wp$ $\mathop B\limits_\wp$ 是论域 $U$ 上的云子集，其隶属函数的云模型数字特征分别为 ${C_A}\left( {{\rm{E}}{x_A},{\rm{E}}{n_A},H{e_A}} \right)$ ${C_B}\left( {{\rm{E}}{x_B},{\rm{E}}{n_B},H{e_B}} \right)$ $U$ 的全体云子集构成一个集合族，称为 $U$ 的云幂集，记为 $\prod {\left( U \right)}$ 。若 $\mathop A\limits_\wp ,\mathop B\limits_\wp \in \prod {\left( U \right)}$ $\mathop A\limits_\wp \cap \mathop B\limits_\wp ,$ $\mathop A\limits_\wp \cup \mathop B\limits_\wp ,\mathop {{A^{\rm{c}}}}\limits_\wp$ 分别表示 $\mathop A\limits_\wp$ $\mathop B\limits_\wp$ 的并集、交集和 $\mathop A\limits_\wp$ 的余(补)集，则云集合的相等关系、包含关系和并集、交集、余集运算方法分别如下。

2.2.1 相等关系

 $\begin{array}{l} \mathop A\limits_\wp = \mathop B\limits_\wp \Leftrightarrow {C_A}({\rm{E}}{x_A},{\rm{E}}{n_A},H{e_A}) = \\ {C_B}({\rm{E}}{x_B},{\rm{E}}{n_B},H{e_B}) \Leftrightarrow \left\{ {\begin{array}{*{20}{c}} {{\rm{E}}{x_A} = {\rm{E}}{x_B}} \\ {{\rm{E}}{n_{\rm{A}}} = {\rm{E}}{n_{\rm{B}}}} \\ {H{e_A} = H{e_B}} \end{array}} \right. \end{array}$

 $\left\{ {\begin{array}{*{20}{l}} {\mathop A\limits_\wp = \mathop B\limits_\wp \Leftrightarrow P(I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )) > I({u_{\mathop B\limits_\wp }}(\mathop x\limits^\wp ))) = \dfrac{1}{2}} \\ {{a^l} = {b^l},{a^r} = {b^r}} \end{array}} \right.$
2.2.2 包含关系

 Download: 图 2 完全包含与不完全包含关系 Fig. 2 Relations between complete and incomplete inclusion

 $\begin{array}{c} \mathop A\limits_\wp \supset \mathop B\limits_\wp \Leftrightarrow P(I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )) > I({u_{\mathop B\limits_\wp }}(\mathop x\limits^\wp ))) = 1 \Leftrightarrow \\ P(I({u_{\mathop B\limits_\wp }}(\mathop x\limits^\wp )) >I( {u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp ))) = 0 \end{array}$

 $\mathop A\limits_\wp \mathop \supseteq \limits^p \mathop B\limits_\wp \Leftrightarrow \frac{{P(I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )) > I({u_{\mathop B\limits_\wp }}(\mathop x\limits^\wp )))}}{{P(I({u_{\mathop B\limits_\wp }}(\mathop x\limits^\wp )) > I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )))}} \geqslant 1$
2.2.3 并集

 ${u_{\mathop A\limits_\wp \cup \mathop B\limits_\wp }}\left( {\mathop x\limits^\wp } \right) = {u_{\mathop A\limits_\wp }}\left( {\mathop x\limits^\wp } \right) \vee {u_{\mathop B\limits_\wp }}\left( {\mathop x\limits^\wp } \right)$

 $\begin{array}{c} {u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp ) \vee {u_{\mathop B\limits_\wp }}(\mathop x\limits^\wp ) = \\ \left\{ {\begin{array}{*{20}{l}} {{u_{\mathop A\limits_\wp }}\left( {\mathop x\limits^\wp } \right)},\quad{\dfrac{{P(I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )) > I({u_{\mathop B\limits_\wp }}(\mathop x\limits^\wp )))}}{{P(I({u_{\mathop B\limits_\wp }}(\mathop x\limits^\wp )) > I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )))}} > 1}\\ {{u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp ),{u_{\mathop B\limits_\wp }}(\mathop x\limits^\wp )},\quad{\dfrac{{P(I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )) > I({u_{\mathop B\limits_\wp }}(\mathop x\limits^\wp )))}}{{P(I({u_{\mathop B\limits_\wp }}(\mathop x\limits^\wp )) > I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )))}} = 1}\\ {{u_{\mathop B\limits_\wp }}(\mathop x\limits^\wp )},\quad{\dfrac{{P(I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )) > I({u_{\mathop B\limits_\wp }}(\mathop x\limits^\wp )))}}{{P(I({u_{\mathop B\limits_\wp }}(\mathop x\limits^\wp )) > I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )))}} <1}\\ {{u_{\mathop A\limits_\wp }}\left( {\mathop x\limits^\wp } \right)},\quad{P(I({u_{\mathop B\limits_\wp }}(\mathop x\limits^\wp )) > I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp ))) = 0} \end{array}} \right. \end{array}$ (2)

 Download: 图 3 两个云集合及其并集隶属函数 Fig. 3 Two cloud sets and the subjection degree function of their union
2.2.4 交集

 ${u_{\mathop A\limits_\wp \cap \mathop B\limits_\wp }}(\mathop x\limits^\wp ) = {u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp ) \wedge {u_{\mathop B\limits_\wp }}\left( {\mathop x\limits^\wp } \right)$

 $\begin{array}{c} {u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp ) \wedge {u_{\mathop B\limits_\wp }}(\mathop x\limits^\wp ) = \\ \left\{ {\begin{array}{*{20}{l}} {{u_{\mathop A\limits_\wp }}\left( {\mathop x\limits^\wp } \right)},\quad{\dfrac{{P(I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )) > I({u_{\mathop B\limits_\wp }}(\mathop x\limits^\wp )))}}{{P(I({u_{\mathop B\limits_\wp }}(\mathop x\limits^\wp )) > I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )))}} <1}\\ {{u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp ),{u_{\mathop B\limits_\wp }}(\mathop x\limits^\wp )},\quad{\dfrac{{P(I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )) > I({u_{\mathop B\limits_\wp }}(\mathop x\limits^\wp )))}}{{P(I({u_{\mathop B\limits_\wp }}(\mathop x\limits^\wp )) > I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )))}} = 1}\\ {{u_{\mathop B\limits_\wp }}(\mathop x\limits^\wp )},\quad{\dfrac{{P(I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )) > I({u_{\mathop B\limits_\wp }}(\mathop x\limits^\wp )))}}{{P(I({u_{\mathop B\limits_\wp }}(\mathop x\limits^\wp )) > I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )))}} > 1}\\ {{u_{\mathop B\limits_\wp }}(\mathop x\limits^\wp )},\quad{P(I({u_{\mathop B\limits_\wp }}(\mathop x\limits^\wp )) > I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp ))) = 0} \end{array}} \right. \end{array}$ (3)

 Download: 图 4 两个云集合的交集隶属函数 Fig. 4 Subjection degree function of the intersection of two cloud sets
2.2.5 余(补)集

 ${u_{{{\mathop A\limits_\wp }^c}}}(\mathop x\limits^\wp ) = 1 - {u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )$

2.2.6 云集合的直积

 ${u_{\mathop A\limits_\wp {\rm{(}}X{\rm{)}} \times \mathop B\limits_\wp {\rm{(}}Y{\rm{)}}}} = {u_{\mathop A\limits_\wp {\rm{(}}X{\rm{)}}}} \wedge {u_{\mathop B\limits_\wp {\rm{(}}Y{\rm{)}}}}$
3 云集合的截集与分解定理 3.1 云集合的截集 3.1.1 云集合的 $\lambda$ 截集

$\mathop A\limits_\wp$ 为论域 $U$ 上的云子集，具有云模型 $C({\rm{E}}x,{\rm{E}}n,He)$ 表示的隶属函数 $u_{\mathop A\limits_\wp }(\mathop x\limits^\wp )$ ，对于任意实数 $\lambda \in [0,1]$ ，则云集合的 $\lambda$ 云截集为

 $({{\mathop A\limits_\wp} _\lambda })(\mathop x\limits^\wp ) = \left\{ {\begin{array}{*{20}{l}} {{u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp ),\quad I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp ))\mathop \geqslant \limits^{\rm{L}} \lambda } \\ {0,\quad I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp ))\neg {\mathop \geqslant \limits^{\rm{L}}} \lambda } \end{array}} \right.$

 $({A_\lambda })(\mathop x\limits^\wp ) = \left\{ {\begin{array}{*{20}{l}} {1,\quad I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp ))\mathop \geqslant \limits^{\rm{L}} \lambda } \\ {0,\quad I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp ))\neg \mathop \geqslant \limits^{\rm{L}} \lambda } \end{array}} \right.$

 $({{\mathop A\limits_{\sim}} _\lambda })(\mathop x\limits^{\sim} ) = \left\{ {\begin{array}{*{20}{l}} {{u_{\mathop A\limits_{\sim} }}(\mathop x\limits^{\sim} ),\quad I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp ))\mathop \geqslant \limits^{\rm{L}} \lambda } \\ {0,\quad I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp ))\neg \mathop \geqslant \limits^{\rm{L}} \lambda } \end{array}} \right.$

 ${{\rm{e}}^{ - \frac{{{{\left( {x - {\rm{E}}x} \right)}^2}}}{{2{{({\rm{E}}n')}^2}}}}} \geqslant \lambda$

 ${A_\lambda } = \left\{ {\left. x \right|{\rm{E}}x{\rm{ - }}\sqrt {{\rm{ - 2}}{\rm{E}}{{n'}^2}\ln \lambda } \leqslant x \leqslant {\rm{E}}x + \sqrt {{\rm{ - 2}}{\rm{E}}{{n'}^2}\ln \lambda } } \right\}$

$({{\mathop {A_\lambda}\limits_ \sim} })(x)$ $({A_\lambda })(x)$ 体现了云集合与模糊集、经典集合之间的关系， $({{\mathop{ A_\lambda }\limits_ \sim} })(x)$ 模糊截集是沟通云集合与模糊集的桥梁，能够实现云集合与模糊集之间的转化； $({A_\lambda })(x)$ 经典截集是沟通云集合与经典集合的桥梁，能够实现云集合与经典集之间的转换。

3.1.2 云集合的元素截集

$\mathop A\limits_\wp$ 为论域 $U$ 上的云子集，具有云模型 $C({\rm{E}}x,{\rm{E}}n,He)$ 表示的隶属函数 $u_{\mathop A\limits_\wp }(\mathop x\limits^\wp )$ ，对于任意集合元素 ${{\mathop x\limits^\wp} _i} \in U$ ，记：

 $({{\mathop A\limits_\wp} _{{{\mathop x\limits^\wp }_i}}})(\mathop x\limits^\wp ) = \left\{ {\begin{array}{*{20}{c}} {{u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp ),\quad \dfrac{{P(I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )) > I({u_{\mathop A\limits_\wp }}(\mathop {{x_i}}\limits^\wp )))}}{{P(I({u_{\mathop A\limits_\wp }}(\mathop {{x_i}}\limits^\wp )) > I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )))}} \geqslant 1} \\ \begin{array}{l} 0,\quad \dfrac{{P(I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )) > I({u_{\mathop A\limits_\wp }}({{\mathop x\limits^\wp }_i})))}}{{P(I({u_{\mathop A\limits_\wp }}({x_i})) > I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp ))}} < 1 \\ {u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp ),\quad P(I({u_{\mathop A\limits_\wp }}({{\mathop x\limits^\wp} _i})) > I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp ))) = 0 \end{array} \end{array}} \right.$
 $({A_{{{\mathop {\rm{x}}\limits^\wp }_i}}})(x) = \left\{ {\begin{array}{*{20}{l}} {1,\quad \dfrac{{P(I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )) > I({u_{\mathop A\limits_\wp }}({{\mathop x\limits^\wp }_i})))}}{{P(I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )) > I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )))}} \geqslant 1} \\ \begin{array}{l} 0,\quad \dfrac{{P(I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )) > I({u_{\mathop A\limits_\wp }}({{\mathop x\limits^\wp }_i})))}}{{P(I({u_{\mathop A\limits_\wp }}({{\mathop x\limits^\wp }_i})) > I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )))}} < 1 \\ 1,\quad P(I({u_{\mathop A\limits_\wp }}({{\mathop x\limits^\wp} _i})) > I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp ))) = 0 \end{array} \end{array}} \right.$ (4)

3.1.3 云集合的区间截集

$\mathop A\limits_\wp$ 为论域 $U$ 上的云子集，具有云模型 $C({\rm{E}}x,{\rm{E}}n,He)$ 表示的隶属函数 $u_{\mathop A\limits_\wp }(\mathop x\limits^\wp )$ ，对于给定的区间 $\forall [a,b] \subseteq [0,1]$ ，记：

 ${{\mathop A\limits_\wp} _{[a,b]}}\left( {\mathop x\limits^\wp } \right) = \left\{ {\begin{array}{*{20}{c}} {{u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp ),\quad \dfrac{{P(I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )) > I([a,b]))}}{{P(I([a,b])) > I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )))}} \geqslant 1}\\ \begin{array}{l} 0,\quad \dfrac{{P(I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )) > I([a,b]))}}{{P(I([a,b])) > I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )))}} < 1\\ {u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp ),\quad P(I([a,b]) > I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp ))) = 0 \end{array} \end{array}} \right.$
 ${A_{[a,b]}}\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {1,\quad \dfrac{{P(I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )) > I([a,b]))}}{{P(I([a,b])) > I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )))}} \geqslant 1}\\ \begin{array}{l} 0,\quad \dfrac{{P(I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )) > I([a,b])}}{{P(I([a,b])) > I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )))}} < 1\\ 1,\quad P(I([a,b]) > I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp ))) = 0 \end{array} \end{array}} \right.$

3.1.4 云集合的随机隶属度截集

$\mathop A\limits_\wp$ $\mathop B\limits_\wp$ 为论域 $U$ 上的云子集，具有云模型 ${C_A}({\rm{E}}x,{\rm{E}}n,He)$ ${C_B}({\rm{E}}x,{\rm{E}}n,He)$ 表示的隶属函数 $u_{\mathop A\limits_\wp }(\mathop x\limits^\wp )$ $u_{\mathop B\limits_\wp }(\mathop x\limits^\wp )$ ，对于任意随机隶属度 $u_{\mathop B\limits_\sim A }(\mathop {{x_{\rm{i}}}}\limits^\wp )$ ( ${{\mathop x\limits^\wp} _i} \in U$ )有

 ${{\mathop A\limits_\wp} _{{u_{\mathop B\limits_\wp }}({{\mathop x\limits^\wp }_{\rm{i}}})}}\left( {\mathop x\limits^\wp } \right) = \left\{ {\begin{array}{*{20}{l}} {{u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp ),\quad \dfrac{{P(I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )) > I({u_{\mathop B\limits_\wp }}({{\mathop x\limits^\wp }_i})))}}{{P(I({u_{\mathop B\limits_\wp }}({{\mathop x\limits^\wp }_i})) > I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )))}} \geqslant 1}\\ \begin{array}{l} 0,\quad \dfrac{{P(I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )) > I({u_{\mathop B\limits_\wp }}({{\mathop x\limits^\wp }_i})))}}{{P(I({u_{\mathop B\limits_\wp }}({{\mathop x\limits^\wp }_i})) > I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )))}} < 1\\ {u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp ),\quad P(I({u_{\mathop B\limits_\wp }}({{\mathop x\limits^\wp} _i})) > I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp ))) = 0 \end{array} \end{array}} \right.$
 ${{\mathop A\limits_{}} _{{u_{\mathop B\limits_\wp }}({{\mathop x\limits^\wp }_i})}}\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {1,\quad \dfrac{{P(I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )) > I({u_{\mathop B\limits_\wp }}({{\mathop x\limits^\wp }_i})))}}{{P(I({u_{\mathop B\limits_\wp }}({{\mathop x\limits^\wp }_i})) > I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )))}} \geqslant 1}\\ \begin{array}{l} 0,\quad \dfrac{{P(I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )) > I({u_{\mathop B\limits_\wp }}({{\mathop x\limits^\wp }_i})))}}{{P(I({u_{\mathop B\limits_\wp }}({{\mathop x\limits^\wp }_i})) > I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )))}} < 1\\ 1,\quad P(I({u_{\mathop B\limits_\wp }}({{\mathop x\limits^\wp} _i})) > I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp ))) = 0 \end{array} \end{array}} \right.$ (5)

${{\mathop A\limits_\wp} _{{u_{\mathop B\limits_\wp }}({{\mathop x\limits^\wp }_i})}}\left( {\mathop x\limits^\wp } \right)$ 为云集合 $\mathop A\limits_\wp$ 的随机隶属度 $u_{\mathop B\limits_\wp }(\mathop x\limits^\wp )$ 云截集，称 ${{{\mathop A\limits_\wp}} _{{u_{\mathop B\limits_\wp }}({{\mathop x\limits^\wp }_i})}}\left( x \right)$ 为云集合 $\mathop A\limits_\wp$ 的随机隶属度 $u_{\mathop B\limits_\wp }(\mathop x\limits^\wp )$ 经典截集。特殊情况下 $\mathop A\limits_\wp$ $\mathop B\limits_\wp$ 可以是同一集合。元素截集与随机隶属度截集本质上是相同的。

3.2 云集合的分解定理

 $\mathop A\limits_\wp = \bigcup\limits_{x \in U}^{} {{u_{\mathop A\limits_\wp }}({{\mathop x\limits^\wp }_i}){A_{{u_{\mathop A\limits_\wp }}({{\mathop x\limits^\wp }_i})}}} \left( x \right)$ (6)

 ${u_{\mathop A\limits_\wp }}({{\mathop x\limits^\wp} _i})A = \left\{ {\begin{array}{*{20}{c}} {{u_{\mathop A\limits_\wp }}({{\mathop x\limits^\wp }_i}),\quad x \in A} \\ {0,\quad x \notin A} \end{array}} \right.$

 $\begin{array}{c} \left( {{u_{\mathop A\limits_\wp }}({{\mathop x\limits^\wp }_i}){A_{_{\mathop A\limits_\wp }({{\mathop x\limits^\wp }_i})}}\left( x \right)} \right)\left( {\mathop x\limits^\wp } \right) = \\ \left\{ {\begin{array}{*{20}{c}} {{u_{\mathop A\limits_\wp }}({{\mathop x\limits^\wp }_i}),\quad\mathop x\limits^\wp \in {A_{{u_{\mathop A\limits_\wp }}({{\mathop x\limits^\wp }_{\rm{i}}})}},\quad \dfrac{{P(I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )) > I({u_{\mathop A\limits_\wp }}({{\mathop x\limits^\wp }_i})))}}{{P(I({u_{\mathop A\limits_\wp }}({{\mathop x\limits^\wp }_i})) > I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )))}} \geqslant 1}\\ \begin{array}{l} 0,\quad \mathop x\limits^\wp \notin {A_{{u_{\mathop A\limits_\wp }}({{\mathop x\limits^\wp }_{\rm{i}}})}},\dfrac{{P(I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )) > I({u_{\mathop A\limits_\wp }}({{\mathop x\limits^\wp }_i})))}}{{P(I({u_{\mathop A\limits_\wp }}({{\mathop x\limits^\wp }_i})) > I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp )))}} < 1\\ {u_{\mathop A\limits_\wp }}({\mathop x\limits^\wp _i}),\quad \mathop x\limits^\wp \in {A_{{u_{\mathop A\limits_\wp }}\left( {{{\mathop x\limits^\wp }_i}} \right)}},P(I({u_{\mathop A\limits_\wp }}({{\mathop x\limits^\wp} _i})) > I({u_{\mathop A\limits_\wp }}(\mathop x\limits^\wp ))) = 0 \end{array} \end{array}} \right. \end{array}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!$ (7)

 $\mathop A\limits_\wp = \bigcup\limits_{{{\mathop x\limits^\wp }_i} \in U} {{u_{\mathop A\limits_\wp }}({{\mathop x\limits^\wp }_i}){A_{{{\mathop x\limits^\wp }_i}}}} (x)$ (8)

 ${A_{{{\mathop x\limits^\wp }_i}}}(x){\rm{ = }}{A_{{u_{\mathop A\limits_\wp }}({{\mathop x\limits^\wp }_i})}}\left( x \right)$

 $\bigcup\limits_{{{\mathop x\limits^\wp }_i} \in U} {{u_{\mathop A\limits_\wp }}({{\mathop x\limits^\wp }_i}){A_{{{\mathop x\limits^\wp }_i}}}} (x){\rm{ = }}\bigcup\limits_{{{\mathop x\limits^\wp }_i} \in U} {{u_{\mathop A\limits_\wp }}({{\mathop x\limits^\wp }_i}){A_{{u_{\mathop A\limits_\wp }}({{\mathop x\limits^\wp }_i})}}\left( x \right)}$

 $\bigcup\limits_{{{\mathop x\limits^\wp }_i} \in U} {{u_{\mathop A\limits_\wp }}({{\mathop x\limits^\wp }_i}){A_{{{\mathop x\limits^\wp }_i}}}} (x) = \mathop A\limits_\wp$
4 结束语

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