Dynamic Fuzzy Machine Learning

(8)Dualization law$\begin{array}{l}{\left(\left(\overleftarrow{A},\overrightarrow{A}\right)\cup \left(\overleftarrow{B},\overrightarrow{B}\right)\right)}^C={\left(\overleftarrow{A},\overrightarrow{A}\right)}^C\cup {\left(\overleftarrow{B},\overrightarrow{B}\right)}^C\\ {\left(\left(\overleftarrow{A},\overrightarrow{A}\right)\cap \left(\overleftarrow{B},\overrightarrow{B}\right)\right)}^C={\left(\overleftarrow{A},\overrightarrow{A}\right)}^C\cup {\left(\overleftarrow{B},\overrightarrow{B}\right)}^C\end{array}$

DF set does not satisfy the complementary law. The fundamental reason is that there is no clear boundary of DF set, that is, the boundary of the dynamic fuzzy.

8.1.3Cut set of dynamic fuzzy sets

In the transformation of DF set and common set, an important concept is the horizontal section set $\left(\overleftarrow{\lambda },\overrightarrow{\lambda }\right).$

Definition 8.4 Assume$\left(\overleftarrow{A},\overrightarrow{A}\right)\in DF\left(U\right),\left(\overleftarrow{\lambda },\overrightarrow{\lambda }\right)\in \left[0,1\right]\times \left[\leftarrow ,\to \right],$ we note the following:

$\begin{array}{l}\left(1\right){\overleftarrow{A}}_{\overleftarrow{A}}=\left\{\left|\overleftarrow{u}\right|\overleftarrow{u}\in U,\overleftarrow{A}\left(\overleftarrow{u}\right)\ge \overleftarrow{\lambda }\right\}\text{or}{\overleftarrow{A}}_{\overleftarrow{\lambda }}=\{\stackrel{}{u}\end{array}$