Dynamic Fuzzy Machine Learning

Definition 8.17 A variable of DF proposition is the variable of closed interval [0, 1] × [←, →]. We call that as DF proposition variable. We usually use lower case letters to express this.

For DF variable $\left(\overleftarrow{x},\overrightarrow{x}\right),\left(\overleftarrow{y},\overrightarrow{y}\right)\in \left[0,1\right]\times \left[\leftarrow ,\to \right],$ the rules are as follows: (notes: $\left(\overleftarrow{x},\overrightarrow{x}\right)=\overleftarrow{x}\text{or}\overrightarrow{x},\max \left(\overleftarrow{x},\overrightarrow{x}\right)\triangleq \overrightarrow{x},\min \left(\overleftarrow{x},\overrightarrow{x}\right)\triangleq \overleftarrow{x})$

(1)Negation “—”.For example: The negation of $\left(\overleftarrow{x},\overrightarrow{x}\right)\text{is}\overline{\left(\overleftarrow{x},\overrightarrow{x}\right)},\mathrm{and}\overline{\left(\overleftarrow{x},\overrightarrow{x}\right)}\triangleq \left(\left(\overleftarrow{1}-\overleftarrow{x}\right),\left(\overrightarrow{1}-\overrightarrow{x}\right)\right);$

(2)Disjunction “ V ”.For example: The disjunction of $\left(\overleftarrow{x},\overrightarrow{x}\right)\mathrm{and}\left(\overleftarrow{y},\overrightarrow{y}\right)\text{is}\left(\overleftarrow{x},\overrightarrow{x}\right)\vee \left(\overleftarrow{y},\overrightarrow{y}\right)\triangleq$$\max (\left(\overleftarrow{x},\overrightarrow{x}\right),(\overleftarrow{y},\overrightarrow{y}$