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8Appendix

8.1Dynamic fuzzy sets

8.1.1Definition of dynamic fuzzy sets

Definition 8.1 A map defined in the domain of discourse $U . (\overset{\leftarrow}{A,} \vec{A}) : (\overset{\leftarrow}{U,} \vec{U}) \to [0, 1] \times$ $U . (\overset{\leftarrow}{A,} \vec{A}) : (\overset{\leftarrow}{U,} \vec{U}) \to [0, 1] \times$ $[\leftarrow, \to], (\overset{\leftarrow}{u}, \vec{u}) \mapsto (\overset{\leftarrow}{A}, (\vec{u}), \overset{\leftarrow}{A}, (\vec{u})),$ $[\leftarrow, \to], (\overset{\leftarrow}{u}, \vec{u}) \mapsto (\overset{\leftarrow}{A}, (\vec{u}), \overset{\leftarrow}{A}, (\vec{u})),$ , marked as $(\overset{\leftarrow}{A}, \vec{A}) = \overset{\leftarrow}{A} or \vec{A} .$ $(\overset{\leftarrow}{A}, \vec{A}) = \overset{\leftarrow}{A} or \vec{A} .$ Then we call $(\overset{\leftarrow}{A}, \vec{A})$ $(\overset{\leftarrow}{A}, \vec{A})$ as the dynamic fuzzy sets in $(\overset{\leftarrow}{U}, \vec{U}),$ $(\overset{\leftarrow}{U}, \vec{U}),$ , shorten for DFS. We call $(\overset{\leftarrow}{A}, (\vec{u}), \overset{\leftarrow}{A}, (\vec{u}))$ $(\overset{\leftarrow}{A}, (\vec{u}), \overset{\leftarrow}{A}, (\vec{u}))$ as the membership function of membership degree $(\overset{\leftarrow}{A}, \vec{A}) .$ $(\overset{\leftarrow}{A}, \vec{A}) .$

*Explanation: For any a ∈ [0, 1], we can make it dynamic fuzzilization as $a \overset{D F}{=} (\overset{\leftarrow}{a}, \vec{a}),$ $a \overset{D F}{=} (\overset{\leftarrow}{a}, \vec{a}),$ $\begin{array}{l} a \overset{D F}{=} \overset{\leftarrow}{a} or \vec{a}, \max (\overset{\leftarrow}{a}, \vec{a}) \overset{Δ}{=} \vec{a}, \min (\overset{\leftarrow}{a}, \vec{a}) \overset{Δ}{=} \overset{\leftarrow}{a} . \end{array}$ $\begin{array}{l} a \overset{D F}{=} \overset{\leftarrow}{a} or \vec{a}, \max (\overset{\leftarrow}{a}, \vec{a}) \overset{Δ}{=} \vec{a}, \min (\overset{\leftarrow}{a}, \vec{a}) \overset{Δ}{=} \overset{\leftarrow}{a} . \end{array}$ So that we can visually present variation trends of number a.

In the domain of discourse U, there are many dynamic fuzzy sets. We mark the entire dynamic fuzzy sets as DF(U), namely:

$\begin{array}{l} D F (U) = {(\overset{\leftarrow}{A}, \vec{A}) | (\overset{\leftarrow}{A}, \vec{A}), (\overset{\leftarrow}{u}, \vec{u}) \mapsto [0, 1] \times [\leftarrow, \to]} \\ = {(A \times (\leftarrow, \to)) | (A \times (\leftarrow, \to)), (u \times (\leftarrow, \to)) \mapsto [0, 1] \times [\leftarrow, \to]} . \end{array}$ $\begin{array}{l} D F (U) = {(\overset{\leftarrow}{A}, \vec{A}) | (\overset{\leftarrow}{A}, \vec{A}), (\overset{\leftarrow}{u}, \vec{u}) \mapsto [0, 1] \times [\leftarrow, \to]} \\ = {(A \times (\leftarrow, \to)) | (A \times (\leftarrow, \to)), (u \times (\leftarrow, \to)) \mapsto [0, 1] \times [\leftarrow, \to]} . \end{array}$

If domain of discourse U is time varied domain, then we define it as UT (there T means time). Correspondingly, we note DF(UT) as

$\begin{array}{l} D F (U_{T}) = {({\overset{\leftarrow}{A}}_{t}, \vec{A_{t}}) | (\overset{\leftarrow}{A_{t}}, \vec{A_{t}}), (\overset{\leftarrow}{u}, \vec{u}) \mapsto [0, 1] \times [\leftarrow, \to]} \\ = {(A_{t} \times (\leftarrow, \to)) | (A_{t} \times (\leftarrow, \to)), (u \times (\leftarrow, \to)) \mapsto [0, 1] \times [\leftarrow, \to]} . \end{array}$ $\begin{array}{l} D F (U_{T}) = {({\overset{\leftarrow}{A}}_{t}, \vec{A_{t}}) | (\overset{\leftarrow}{A_{t}}, \vec{A_{t}}), (\overset{\leftarrow}{u}, \vec{u}) \mapsto [0, 1] \times [\leftarrow, \to]} \\ = {(A_{t} \times (\leftarrow, \to)) | (A_{t} \times (\leftarrow, \to)), (u \times (\leftarrow, \to)) \mapsto [0, 1] \times [\leftarrow, \to]} . \end{array}$

(there t ∈ T).

8.1.2Operation of dynamic fuzzy sets

Operations between two DF subsets can be understood as the corresponding operation of membership function, we have the following definition: “ ∀ ” means “any”, “ ” means “exist”.

Definition 8.2 Assume $\begin{array}{l} (\overset{\leftarrow}{A}, \vec{A}) and (\overset{\leftarrow}{B}, \vec{B}) \in D F (U), if \forall (\overset{\leftarrow}{u}, \vec{u}) \in U and \end{array}$ $\begin{array}{l} (\overset{\leftarrow}{A}, \vec{A}) and (\overset{\leftarrow}{B}, \vec{B}) \in D F (U), if \forall (\overset{\leftarrow}{u}, \vec{u}) \in U and \end{array}$ $(\overset{\leftarrow}{B}, \vec{B}) (\overset{\leftarrow}{u}, \vec{u}) and (\overset{\leftarrow}{A}, \vec{A}) (\overset{\leftarrow}{u}, \vec{u}), then (\overset{\leftarrow}{A}, \vec{A}) contains (\overset{\leftarrow}{B}, \vec{B}), we note that (\overset{\leftarrow}{B}, \vec{B}) \subseteq$ $(\overset{\leftarrow}{B}, \vec{B}) (\overset{\leftarrow}{u}, \vec{u}) and (\overset{\leftarrow}{A}, \vec{A}) (\overset{\leftarrow}{u}, \vec{u}), then (\overset{\leftarrow}{A}, \vec{A}) contains (\overset{\leftarrow}{B}, \vec{B}), we note that (\overset{\leftarrow}{B}, \vec{B}) \subseteq$ $(\overset{\leftarrow}{A}, \vec{A});$ $(\overset{\leftarrow}{A}, \vec{A});$

$I f (\overset{\leftarrow}{A}, \vec{A}) \subseteq (\overset{\leftarrow}{B}, \vec{B}), and (\overset{\leftarrow}{B}, \vec{B}) \subseteq (\overset{\leftarrow}{A}, \vec{A}), t h e n (\overset{\leftarrow}{A}, \vec{A}) = (\overset{\leftarrow}{B}, \vec{B}) .$ $I f (\overset{\leftarrow}{A}, \vec{A}) \subseteq (\overset{\leftarrow}{B}, \vec{B}), and (\overset{\leftarrow}{B}, \vec{B}) \subseteq (\overset{\leftarrow}{A}, \vec{A}), t h e n (\overset{\leftarrow}{A}, \vec{A}) = (\overset{\leftarrow}{B}, \vec{B}) .$

Obviously, contain relationship “ ⊆ ” is a kind of relationship on DF power set U, with the follow property:

(1)Reflexivity
$\begin{array}{l} \forall (\overset{\leftarrow}{A}, \vec{A}) \in D F (U), (\overset{\leftarrow}{A}, \vec{A}) \subseteq (\overset{\leftarrow}{A}, \vec{A}); \end{array}$ $\begin{array}{l} \forall (\overset{\leftarrow}{A}, \vec{A}) \in D F (U), (\overset{\leftarrow}{A}, \vec{A}) \subseteq (\overset{\leftarrow}{A}, \vec{A}); \end{array}$

(2)Anti-symmetry If
$\begin{array}{l} If (\overset{\leftarrow}{A}, \vec{A}) \subseteq (\overset{\leftarrow}{B}, \vec{B}), (\overset{\leftarrow}{B}, \vec{B}) \subseteq (\overset{\leftarrow}{A}, \vec{A}) \Rightarrow (\overset{\leftarrow}{A}, \vec{A}) = (\overset{\leftarrow}{B}, \vec{B}); \end{array}$ $\begin{array}{l} If (\overset{\leftarrow}{A}, \vec{A}) \subseteq (\overset{\leftarrow}{B}, \vec{B}), (\overset{\leftarrow}{B}, \vec{B}) \subseteq (\overset{\leftarrow}{A}, \vec{A}) \Rightarrow (\overset{\leftarrow}{A}, \vec{A}) = (\overset{\leftarrow}{B}, \vec{B}); \end{array}$

(3)Transitivity If
$\begin{array}{l} If (\overset{\leftarrow}{A}, \vec{A}) \subseteq (\overset{\leftarrow}{B}, \vec{B}), (\overset{\leftarrow}{B}, \vec{B}) \subseteq (\overset{\leftarrow}{C}, \vec{C}) \Rightarrow (\overset{\leftarrow}{A}, \vec{A}) \subseteq (\overset{\leftarrow}{C}, \vec{C}) . \end{array}$ $\begin{array}{l} If (\overset{\leftarrow}{A}, \vec{A}) \subseteq (\overset{\leftarrow}{B}, \vec{B}), (\overset{\leftarrow}{B}, \vec{B}) \subseteq (\overset{\leftarrow}{C}, \vec{C}) \Rightarrow (\overset{\leftarrow}{A}, \vec{A}) \subseteq (\overset{\leftarrow}{C}, \vec{C}) . \end{array}$

Definition 8.3 Assume $\begin{array}{l} (\overset{\leftarrow}{A}, \vec{A}), (\overset{\leftarrow}{B}, \vec{B}) \in D F (U), \end{array}$ $\begin{array}{l} (\overset{\leftarrow}{A}, \vec{A}), (\overset{\leftarrow}{B}, \vec{B}) \in D F (U), \end{array}$ we refer the operation $(\overset{\leftarrow}{A}, \vec{A}) \cup$ $(\overset{\leftarrow}{A}, \vec{A}) \cup$ $\begin{array}{l} (\overset{\leftarrow}{B}, \vec{B}), (\overset{\leftarrow}{A}, \vec{A}) \cap (\overset{\leftarrow}{B}, \vec{B}) \end{array}$ $\begin{array}{l} (\overset{\leftarrow}{B}, \vec{B}), (\overset{\leftarrow}{A}, \vec{A}) \cap (\overset{\leftarrow}{B}, \vec{B}) \end{array}$ as the union and intersection of $(\overset{\leftarrow}{A}, \vec{A}), (\overset{\leftarrow}{B}, \vec{B}) . (\overset{\leftarrow}{A}, \vec{A})^{c}$ $(\overset{\leftarrow}{A}, \vec{A}), (\overset{\leftarrow}{B}, \vec{B}) . (\overset{\leftarrow}{A}, \vec{A})^{c}$ is the complement of $(\overset{\leftarrow}{A}, \vec{A}) .$ $(\overset{\leftarrow}{A}, \vec{A}) .$ Their membership function is

$\begin{array}{l} ((\overset{\leftarrow}{A}, \vec{A}) \cup (\overset{\leftarrow}{B}, \vec{B})) (u) = (\overset{\leftarrow}{A}, \vec{A}) (u) \lor (\overset{\leftarrow}{B}, \vec{B}) (u) \\ \overset{Δ}{=} \max ((\overset{\leftarrow}{A}, \vec{A}) (u), (\overset{\leftarrow}{B}, \vec{B}) (u)) \\ ((\overset{\leftarrow}{A}, \vec{A}) \cap (\overset{\leftarrow}{B}, \vec{B})) (u) = (\overset{\leftarrow}{A}, \vec{A}) (u) \lor (\overset{\leftarrow}{B}, \vec{B}) (u) \\ \overset{Δ}{=} \min ((\overset{\leftarrow}{A}, \vec{A}) (u), (\overset{\leftarrow}{B}, \vec{B}) (u)) \\ {(\overset{\leftarrow}{A}, \vec{A})}^{c} (u) = 1 - (\overset{\leftarrow}{A}, \vec{A}) (u) \\ \overset{Δ}{=} (\overset{\leftarrow}{1} - \overset{\leftarrow}{A} (\overset{\leftarrow}{u}), \vec{1} - \vec{A} (\vec{u})) \end{array}$ $\begin{array}{l} ((\overset{\leftarrow}{A}, \vec{A}) \cup (\overset{\leftarrow}{B}, \vec{B})) (u) = (\overset{\leftarrow}{A}, \vec{A}) (u) \lor (\overset{\leftarrow}{B}, \vec{B}) (u) \\ \overset{Δ}{=} \max ((\overset{\leftarrow}{A}, \vec{A}) (u), (\overset{\leftarrow}{B}, \vec{B}) (u)) \\ ((\overset{\leftarrow}{A}, \vec{A}) \cap (\overset{\leftarrow}{B}, \vec{B})) (u) = (\overset{\leftarrow}{A}, \vec{A}) (u) \lor (\overset{\leftarrow}{B}, \vec{B}) (u) \\ \overset{Δ}{=} \min ((\overset{\leftarrow}{A}, \vec{A}) (u), (\overset{\leftarrow}{B}, \vec{B}) (u)) \\ {(\overset{\leftarrow}{A}, \vec{A})}^{c} (u) = 1 - (\overset{\leftarrow}{A}, \vec{A}) (u) \\ \overset{Δ}{=} (\overset{\leftarrow}{1} - \overset{\leftarrow}{A} (\overset{\leftarrow}{u}), \vec{1} - \vec{A} (\vec{u})) \end{array}$

[where $u \overset{D F}{=} (\overset{\leftarrow}{u}, \vec{u})] .$ $u \overset{D F}{=} (\overset{\leftarrow}{u}, \vec{u})] .$

It divides by whether domain U is finite or infinite.

(1)Domain $U = {(\overset{\leftarrow}{u_{1}}, \vec{u_{1}}), (\overset{\leftarrow}{u_{2}}, \vec{u_{2}}), ......, (\overset{\leftarrow}{u_{N}}, \vec{u_{N}})}$ $U = {(\overset{\leftarrow}{u_{1}}, \vec{u_{1}}), (\overset{\leftarrow}{u_{2}}, \vec{u_{2}}), ......, (\overset{\leftarrow}{u_{N}}, \vec{u_{N}})}$ is finite set and DF set.

$\begin{array}{l} (\overset{\leftarrow}{A}, \vec{A}) = (\sum \frac{\overset{\leftarrow}{A} ({\vec{u}}_{i})}{\overset{\leftarrow}{u_{i}}}, \sum \frac{\vec{A} ({\vec{u}}_{i})}{{\vec{u}}_{i}}) \\ (\overset{\leftarrow}{B}, \vec{B}) = (\sum \frac{\overset{\leftarrow}{B} ({\vec{u}}_{i})}{\overset{\leftarrow}{u_{i}}}, \sum \frac{\vec{B} ({\vec{u}}_{i})}{{\vec{u}}_{i}}) \end{array}$ $\begin{array}{l} (\overset{\leftarrow}{A}, \vec{A}) = (\sum \frac{\overset{\leftarrow}{A} ({\vec{u}}_{i})}{\overset{\leftarrow}{u_{i}}}, \sum \frac{\vec{A} ({\vec{u}}_{i})}{{\vec{u}}_{i}}) \\ (\overset{\leftarrow}{B}, \vec{B}) = (\sum \frac{\overset{\leftarrow}{B} ({\vec{u}}_{i})}{\overset{\leftarrow}{u_{i}}}, \sum \frac{\vec{B} ({\vec{u}}_{i})}{{\vec{u}}_{i}}) \end{array}$

Then,

$\begin{array}{l} (\overset{\leftarrow}{A}, \vec{A}) \cup (\overset{\leftarrow}{B}, \vec{B}) = (Σ \frac{\overset{\leftarrow}{A} (\overset{\leftarrow}{u_{i}}) \lor \overset{\leftarrow}{B} (\overset{\leftarrow}{u_{i}})}{\overset{\leftarrow}{u_{i}}}, Σ \frac{\vec{A} (\vec{u_{i}}) \lor \vec{B} (\vec{u_{i}})}{\vec{u_{i}}}) \\ (\overset{\leftarrow}{A}, \vec{A}) \cup (\overset{\leftarrow}{B}, \vec{B}) = (Σ \frac{\overset{\leftarrow}{A} (\overset{\leftarrow}{u_{i}}) \land \overset{\leftarrow}{B} (\overset{\leftarrow}{u_{i}})}{\overset{\leftarrow}{u_{i}}}, Σ \frac{\vec{A} (\vec{u_{i}}) \land \vec{B} (\vec{u_{i}})}{\vec{u_{i}}}) \\ {(\overset{\leftarrow}{A}, \vec{A})}^{c} = (Σ \frac{\overset{\leftarrow}{1} - \overset{\leftarrow}{A} (\overset{\leftarrow}{u_{i}})}{{\overset{\leftarrow}{u}}_{i}}, Σ \frac{\vec{1} - \vec{A} ({\vec{u}}_{i})}{{\vec{u}}_{i}}) . \end{array}$ $\begin{array}{l} (\overset{\leftarrow}{A}, \vec{A}) \cup (\overset{\leftarrow}{B}, \vec{B}) = (Σ \frac{\overset{\leftarrow}{A} (\overset{\leftarrow}{u_{i}}) \lor \overset{\leftarrow}{B} (\overset{\leftarrow}{u_{i}})}{\overset{\leftarrow}{u_{i}}}, Σ \frac{\vec{A} (\vec{u_{i}}) \lor \vec{B} (\vec{u_{i}})}{\vec{u_{i}}}) \\ (\overset{\leftarrow}{A}, \vec{A}) \cup (\overset{\leftarrow}{B}, \vec{B}) = (Σ \frac{\overset{\leftarrow}{A} (\overset{\leftarrow}{u_{i}}) \land \overset{\leftarrow}{B} (\overset{\leftarrow}{u_{i}})}{\overset{\leftarrow}{u_{i}}}, Σ \frac{\vec{A} (\vec{u_{i}}) \land \vec{B} (\vec{u_{i}})}{\vec{u_{i}}}) \\ {(\overset{\leftarrow}{A}, \vec{A})}^{c} = (Σ \frac{\overset{\leftarrow}{1} - \overset{\leftarrow}{A} (\overset{\leftarrow}{u_{i}})}{{\overset{\leftarrow}{u}}_{i}}, Σ \frac{\vec{1} - \vec{A} ({\vec{u}}_{i})}{{\vec{u}}_{i}}) . \end{array}$

(2)Domain U is infinite set and DF set

$\begin{array}{l} (\overset{\leftarrow}{A}, \vec{A}) = (\int_{\overset{\leftarrow}{u} \in U} \frac{\overset{\leftarrow}{A} (\vec{u})}{\overset{\leftarrow}{u}}, \int_{\vec{u} \in U} \frac{\vec{A} (\vec{u})}{\vec{u}}) \\ (\overset{\leftarrow}{B}, \vec{B}) = (\int_{\overset{\leftarrow}{u} \in U} \frac{\overset{\leftarrow}{B} (\vec{u})}{\overset{\leftarrow}{u}}, \int_{\vec{u} \in U} \frac{\vec{B} (\vec{u})}{\vec{u}}) \end{array}$ $\begin{array}{l} (\overset{\leftarrow}{A}, \vec{A}) = (\int_{\overset{\leftarrow}{u} \in U} \frac{\overset{\leftarrow}{A} (\vec{u})}{\overset{\leftarrow}{u}}, \int_{\vec{u} \in U} \frac{\vec{A} (\vec{u})}{\vec{u}}) \\ (\overset{\leftarrow}{B}, \vec{B}) = (\int_{\overset{\leftarrow}{u} \in U} \frac{\overset{\leftarrow}{B} (\vec{u})}{\overset{\leftarrow}{u}}, \int_{\vec{u} \in U} \frac{\vec{B} (\vec{u})}{\vec{u}}) \end{array}$

Then,

$\begin{array}{l} (\overset{\leftarrow}{A}, \vec{A}) \cup (\overset{\leftarrow}{B}, \vec{B}) = (\int \frac{\overset{\leftarrow}{A} (\vec{u}) \lor \overset{\leftarrow}{B} (\overset{\leftarrow}{u})}{\overset{\leftarrow}{u}}, \int \frac{\vec{A} (\vec{u}) \lor \vec{B} (\vec{u})}{\vec{u}}) \\ (\overset{\leftarrow}{A}, \vec{A}) \cup (\overset{\leftarrow}{B}, \vec{B}) = (\int \frac{\overset{\leftarrow}{A} (\vec{u}) \land \overset{\leftarrow}{B} (\overset{\leftarrow}{u})}{\overset{\leftarrow}{u}}, \int \frac{\vec{A} (\vec{u}) \land \vec{B} (\vec{u})}{\vec{u}}) \\ {(\overset{\leftarrow}{A}, \vec{A})}^{c} = (\int \frac{\overset{\leftarrow}{1} - \overset{\leftarrow}{A} (\overset{\leftarrow}{u})}{\overset{\leftarrow}{u}}, \int \frac{\vec{1} - \vec{A} (\vec{u})}{\vec{u}}) . \end{array}$ $\begin{array}{l} (\overset{\leftarrow}{A}, \vec{A}) \cup (\overset{\leftarrow}{B}, \vec{B}) = (\int \frac{\overset{\leftarrow}{A} (\vec{u}) \lor \overset{\leftarrow}{B} (\overset{\leftarrow}{u})}{\overset{\leftarrow}{u}}, \int \frac{\vec{A} (\vec{u}) \lor \vec{B} (\vec{u})}{\vec{u}}) \\ (\overset{\leftarrow}{A}, \vec{A}) \cup (\overset{\leftarrow}{B}, \vec{B}) = (\int \frac{\overset{\leftarrow}{A} (\vec{u}) \land \overset{\leftarrow}{B} (\overset{\leftarrow}{u})}{\overset{\leftarrow}{u}}, \int \frac{\vec{A} (\vec{u}) \land \vec{B} (\vec{u})}{\vec{u}}) \\ {(\overset{\leftarrow}{A}, \vec{A})}^{c} = (\int \frac{\overset{\leftarrow}{1} - \overset{\leftarrow}{A} (\overset{\leftarrow}{u})}{\overset{\leftarrow}{u}}, \int \frac{\vec{1} - \vec{A} (\vec{u})}{\vec{u}}) . \end{array}$

Theorem 8.1 DF(U), ⋃, ⋂, c) have the follow properties:

(1)Idempotent law
$\begin{array}{l} (\overset{\leftarrow}{A}, \vec{A}) \cup (\overset{\leftarrow}{A}, \vec{A}) = (\overset{\leftarrow}{A}, \vec{A}) \\ (\overset{\leftarrow}{A}, \vec{A}) \cap (\overset{\leftarrow}{A}, \vec{A}) = (\overset{\leftarrow}{A}, \vec{A}) \end{array}$ $\begin{array}{l} (\overset{\leftarrow}{A}, \vec{A}) \cup (\overset{\leftarrow}{A}, \vec{A}) = (\overset{\leftarrow}{A}, \vec{A}) \\ (\overset{\leftarrow}{A}, \vec{A}) \cap (\overset{\leftarrow}{A}, \vec{A}) = (\overset{\leftarrow}{A}, \vec{A}) \end{array}$

(2)Commutative law
$\begin{array}{l} (\overset{\leftarrow}{A}, \vec{A}) \cup (\overset{\leftarrow}{B}, \vec{B}) = (\overset{\leftarrow}{B}, \vec{B}) \cup (\overset{\leftarrow}{A}, \vec{A}) \\ (\overset{\leftarrow}{A}, \vec{A}) \cap (\overset{\leftarrow}{B}, \vec{B}) = (\overset{\leftarrow}{B}, \vec{B}) \cap (\overset{\leftarrow}{A}, \vec{A}) \end{array}$ $\begin{array}{l} (\overset{\leftarrow}{A}, \vec{A}) \cup (\overset{\leftarrow}{B}, \vec{B}) = (\overset{\leftarrow}{B}, \vec{B}) \cup (\overset{\leftarrow}{A}, \vec{A}) \\ (\overset{\leftarrow}{A}, \vec{A}) \cap (\overset{\leftarrow}{B}, \vec{B}) = (\overset{\leftarrow}{B}, \vec{B}) \cap (\overset{\leftarrow}{A}, \vec{A}) \end{array}$

(3)Associative law
$\begin{array}{l} ((\overset{\leftarrow}{A}, \vec{A}) \cup (\overset{\leftarrow}{B}, \vec{B})) \cup (\overset{\leftarrow}{C}, \vec{C}) = (\overset{\leftarrow}{A}, \vec{A}) \cup ((\overset{\leftarrow}{B}, \vec{B}) \cup (\overset{\leftarrow}{C}, \vec{C})) \\ ((\overset{\leftarrow}{A}, \vec{A}) \cap (\overset{\leftarrow}{B}, \vec{B}) \cap (\overset{\leftarrow}{C}, \vec{C}) = (\overset{\leftarrow}{A}, \vec{A}) \cap ((\overset{\leftarrow}{B}, \vec{B}) \cap (\overset{\leftarrow}{C}, \vec{C})) \end{array}$ $\begin{array}{l} ((\overset{\leftarrow}{A}, \vec{A}) \cup (\overset{\leftarrow}{B}, \vec{B})) \cup (\overset{\leftarrow}{C}, \vec{C}) = (\overset{\leftarrow}{A}, \vec{A}) \cup ((\overset{\leftarrow}{B}, \vec{B}) \cup (\overset{\leftarrow}{C}, \vec{C})) \\ ((\overset{\leftarrow}{A}, \vec{A}) \cap (\overset{\leftarrow}{B}, \vec{B}) \cap (\overset{\leftarrow}{C}, \vec{C}) = (\overset{\leftarrow}{A}, \vec{A}) \cap ((\overset{\leftarrow}{B}, \vec{B}) \cap (\overset{\leftarrow}{C}, \vec{C})) \end{array}$

(4)Absorption law
$\begin{array}{l} ((\overset{\leftarrow}{A}, \vec{A}) \cup (\overset{\leftarrow}{B}, \vec{B})) \cap (\overset{\leftarrow}{A}, \vec{A}) = (\overset{\leftarrow}{A}, \vec{A}) \\ ((\overset{\leftarrow}{A}, \vec{A}) \cap (\overset{\leftarrow}{B}, \vec{B})) \cup (\overset{\leftarrow}{A}, \vec{A}) = (\overset{\leftarrow}{A}, \vec{A}) \end{array}$ $\begin{array}{l} ((\overset{\leftarrow}{A}, \vec{A}) \cup (\overset{\leftarrow}{B}, \vec{B})) \cap (\overset{\leftarrow}{A}, \vec{A}) = (\overset{\leftarrow}{A}, \vec{A}) \\ ((\overset{\leftarrow}{A}, \vec{A}) \cap (\overset{\leftarrow}{B}, \vec{B})) \cup (\overset{\leftarrow}{A}, \vec{A}) = (\overset{\leftarrow}{A}, \vec{A}) \end{array}$

(5)Distribution law
$\begin{array}{l} ((\overset{\leftarrow}{A}, \vec{A}) \cup (\overset{\leftarrow}{B}, \vec{B})) \cap (\overset{\leftarrow}{C}, \vec{C}) = ((\overset{\leftarrow}{A}, \vec{A}) \cap ((\overset{\leftarrow}{C}, \vec{C}) \cup ((\overset{\leftarrow}{B}, \vec{B}) \cap (\overset{\leftarrow}{C}, \vec{C})) \\ ((\overset{\leftarrow}{A}, \vec{A}) \cap (\overset{\leftarrow}{B}, \vec{B})) \cup (\overset{\leftarrow}{C}, \vec{C}) = (\overset{\leftarrow}{A}, \vec{A}) \cup ((\overset{\leftarrow}{C}, \vec{C}) \cap (\overset{\leftarrow}{B}, \vec{B})) \cup (\overset{\leftarrow}{C}, \vec{C})) \end{array}$ $\begin{array}{l} ((\overset{\leftarrow}{A}, \vec{A}) \cup (\overset{\leftarrow}{B}, \vec{B})) \cap (\overset{\leftarrow}{C}, \vec{C}) = ((\overset{\leftarrow}{A}, \vec{A}) \cap ((\overset{\leftarrow}{C}, \vec{C}) \cup ((\overset{\leftarrow}{B}, \vec{B}) \cap (\overset{\leftarrow}{C}, \vec{C})) \\ ((\overset{\leftarrow}{A}, \vec{A}) \cap (\overset{\leftarrow}{B}, \vec{B})) \cup (\overset{\leftarrow}{C}, \vec{C}) = (\overset{\leftarrow}{A}, \vec{A}) \cup ((\overset{\leftarrow}{C}, \vec{C}) \cap (\overset{\leftarrow}{B}, \vec{B})) \cup (\overset{\leftarrow}{C}, \vec{C})) \end{array}$

(6)Zero-one law
$\begin{array}{l} (\overset{\leftarrow}{A}, \vec{A}) \cup (\overset{\leftarrow}{ϕ}, \vec{ϕ}) = (\overset{\leftarrow}{A}, \vec{A}) \\ (\overset{\leftarrow}{A}, \vec{A}) \cap (\overset{\leftarrow}{ϕ}, \vec{ϕ}) = (\overset{\leftarrow}{ϕ}, \vec{ϕ}) \\ (\overset{\leftarrow}{A}, \vec{A}) \cup (\overset{\leftarrow}{U}, \vec{U}) = (\overset{\leftarrow}{U}, \vec{U}) \\ (\overset{\leftarrow}{A}, \vec{A}) \cap (\overset{\leftarrow}{U}, \vec{U}) = (\overset{\leftarrow}{A}, \vec{A}) \end{array}$ $\begin{array}{l} (\overset{\leftarrow}{A}, \vec{A}) \cup (\overset{\leftarrow}{ϕ}, \vec{ϕ}) = (\overset{\leftarrow}{A}, \vec{A}) \\ (\overset{\leftarrow}{A}, \vec{A}) \cap (\overset{\leftarrow}{ϕ}, \vec{ϕ}) = (\overset{\leftarrow}{ϕ}, \vec{ϕ}) \\ (\overset{\leftarrow}{A}, \vec{A}) \cup (\overset{\leftarrow}{U}, \vec{U}) = (\overset{\leftarrow}{U}, \vec{U}) \\ (\overset{\leftarrow}{A}, \vec{A}) \cap (\overset{\leftarrow}{U}, \vec{U}) = (\overset{\leftarrow}{A}, \vec{A}) \end{array}$

(7)Pull back law
${({(\overset{\leftarrow}{A}, \vec{A})}^{c})}^{c} = (\overset{\leftarrow}{A}, \vec{A})$ ${({(\overset{\leftarrow}{A}, \vec{A})}^{c})}^{c} = (\overset{\leftarrow}{A}, \vec{A})$

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Table of Contents for 8 Appendix

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8Appendix

8.1Dynamic fuzzy sets

8.1.1Definition of dynamic fuzzy sets

8.1.2Operation of dynamic fuzzy sets

Table of Contents for
8 Appendix