Dynamic Fuzzy Machine Learning

Other forms are similar to prove.

[Prove up].

Theorem 8.7 Assume $\left(\overleftarrow{A},\overrightarrow{A}\right)\mathrm{and}\left(\overleftarrow{B},\overrightarrow{B}\right)\in D\text{F}\left(U\right),$ then the necessary and sufficient condition for $\left(\overleftarrow{A},\overrightarrow{A}\right)\in \left(\overleftarrow{B},\overrightarrow{B}\right)\text{is}\left({\overleftarrow{A}}_{\overleftarrow{\alpha }},{\overrightarrow{A}}_{\overrightarrow{\alpha }}\right)\subset \left({\overleftarrow{B}}_{\overleftarrow{\alpha }},{\overrightarrow{B}}_{\overrightarrow{\alpha }}\right)\left(\left(\overleftarrow{\alpha },\overrightarrow{\alpha }\right)\in \left[0,1\right]\times \left[\leftarrow ,\to \right]\right)$ or $\left({\overleftarrow{A}}_{\underset{\dot{}}{\overleftarrow{\alpha }}},{\overleftarrow{A}}_{\underset{\dot{}}{\overleftarrow{\alpha }}}\right)\subset \left({\overleftarrow{B}}_{\underset{\dot{}}{\overleftarrow{\alpha }}},{\overrightarrow{B}}_{\underset{\dot{}}{\overleftarrow{\alpha }}}\right)\left(\left(\overleftarrow{\alpha },\overrightarrow{\alpha }\right)\in \left[0,1\right]\times \left[\leftarrow ,\to \right]\right)\text{or}\left({\overleftarrow{A}}_{\overleftarrow{\alpha }},{\overleftarrow{A}}_{\overrightarrow{\alpha }}\right)\subset \left({\overleftarrow{B}}_{\overleftarrow{\alpha }},{\overrightarrow{B}}_{\overrightarrow{\alpha }}\right)(\left(\overleftarrow{\alpha },\overrightarrow{\alpha }\right)\in$ R0, where R0 is the rational point set in [0, 1]).

Proof: Obvious necessity.

If $\left({\overleftarrow{A}}_{\overleftarrow{\alpha }},{\overleftarrow{A}}_{\overleftarrow{\alpha }}\right)\subset \left({\overleftarrow{B}}_{\overleftarrow{\alpha }},{\overrightarrow{B}}_{\overleftarrow{\alpha }}\right)\left(\left(\overleftarrow{\alpha },\overrightarrow{\alpha }\right)\in \left[0,1\right]\times \left[\leftarrow ,\to \right]\right),$ according ...