8.1.1Definition of dynamic fuzzy sets
Definition 8.1 A map defined in the domain of discourse U.(A,←−A→):(U,←−U→)→[0,1]×[←,→],(u←,u→)↦(A←,(u→),A←,(u→)),, marked as (A←,A→)=A← or A→. Then we call (A←,A→) as the dynamic fuzzy sets in (U←,U→),, shorten for DFS. We call (A←,(u→),A←,(u→)) as the membership function of membership degree (A←,A→).
*Explanation: For any a ∈ [0, 1], we can make it dynamic fuzzilization as a=DF(a←,a→),a=DFa← or a→, max(a← ,a→)=Δa→,min(a← ,a→)=Δa←. 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:
DF(U)={(A←,A→)∣∣∣(A←,A→),(u←,u→)↦[0,1]×[←,→]} ={(A×(←,→))|(A×(←,→)),(u×(←,→))↦[0,1]×[←,→]}.
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
DF(UT)={(A←t,At−→)∣∣∣(At←−,At−→),(u←,u→)↦[0,1]×[←,→]} ={(At×(←,→))|(At×(←,→)),(u×(←,→))↦[0,1]×[←,→]}.
(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 (A←,A→) and (B←,B→) ∈DF(U), if ∀(u←,u→)∈U and (B←,B→) (u←,u→) and (A←,A→)(u←,u→) , then (A←,A→) contains (B←,B→), we note that (B←,B→)⊆ (A←,A→);
If(A←,A→)⊆(B←,B→),and (B←,B→)⊆(A←,A→),then(A←,A→)=(B←,B→).
Obviously, contain relationship “ ⊆ ” is a kind of relationship on DF power set U, with the follow property:
(1)Reflexivity
∀(A←,A→) ∈DF(U), (A←,A→)⊆ (A←,A→);
(2)Anti-symmetry If
If(A←,A→) ⊆ (B←,B→), (B←,B→) ⊆ (A←,A→)⇒(A←,A→)=(B←,B→);
(3)Transitivity If
If(A←,A→) ⊆ (B←,B→), (B←,B→) ⊆ (C←,C→)⇒(A←,A→)⊆(C←,C→).
Definition 8.3 Assume (A←,A→),(B←,B→) ∈DF(U), we refer the operation (A←,A→) ∪ (B←,B→),(A←,A→) ∩ (B←,B→) as the union and intersection of (A←,A→), (B←,B→).(A←,A→) c is the complement of (A←,A→). Their membership function is
((A←,A→)∪(B←,B→))(u)=(A←,A→)(u)∨(B←,B→)(u) =Δmax((A←,A→)(u),(B←,B→)(u))((A←,A→)∩(B←,B→))(u)=(A←,A→)(u)∨(B←,B→)(u) =Δmin((A←,A→)(u),(B←,B→)(u)) (A←,A→)c(u)=1−(A←,A→)(u) =Δ(1←−A←(u←),1→−A→(u→))
[where u=DF (u←,u→)].
It divides by whether domain U is finite or infinite.
(1)Domain U={(u1←,u1→),(u2←,u2→),......,(uN←−,uN−→)} is finite set and DF set.
(A←,A→)=⎛⎝⎜⎜∑A←(u→i)ui←,∑A→(u→i)u→i⎞⎠⎟⎟(B←,B→)=⎛⎝⎜⎜∑B←(u→i)ui←,∑B→(u→i)u→i⎞⎠⎟⎟
Then,
(A←,A→)∪(B←,B→)=(ΣA←(ui←)∨B←(ui←)ui←,ΣA→(ui→)∨B→(ui→)ui→)(A←,A→)∪(B←,B→)=(ΣA←(ui←)∧B←(ui←)ui←,ΣA→(ui→)∧B→(ui→)ui→) (A←,A→)c=(Σ1←−A←(ui←)u←i,Σ1→−A→(u→i)u→i).
(2)Domain U is infinite set and DF set
(A←,A→)=⎛⎝⎜⎜∫u←∈UA←(u→)u←,∫u→∈UA→(u→)u→⎞⎠⎟⎟(B←,B→)=⎛⎝⎜⎜∫u←∈UB←(u→)u←,∫u→∈UB→(u→)u→⎞⎠⎟⎟
Then,
(A←,A→)∪(B←,B→)=⎛⎝⎜⎜∫A←(u→)∨B←(u←)u←,∫A→(u→)∨B→(u→)u→⎞⎠⎟⎟(A←,A→)∪(B←,B→)=⎛⎝⎜⎜∫A←(u→)∧B←(u←)u←,∫A→(u→)∧B→(u→)u→⎞⎠⎟⎟ (A←,A→)c=⎛⎝⎜⎜∫1←−A←(u←)u←,∫1→−A→(u→)u→⎞⎠⎟⎟.
Theorem 8.1 DF(U), ⋃, ⋂, c) have the follow properties:
(1)Idempotent law
(A←,A→)∪(A←,A→)=(A←,A→)(A←,A→)∩(A←,A→)=(A←,A→)
(2)Commutative law
(A←,A→)∪(B←,B→)=(B←,B→)∪(A←,A→)(A←,A→)∩(B←,B→)=(B←,B→)∩(A←,A→)
(3)Associative law
((A←,A→)∪(B←,B→))∪(C←,C→)=(A←,A→)∪((B←,B→)∪(C←,C→))((A←,A→)∩(B←,B→)∩(C←,C→)=(A←,A→)∩((B←,B→)∩(C←,C→))
(4)Absorption law
((A←,A→)∪(B←,B→))∩(A←,A→)=(A←,A→)((A←,A→)∩(B←,B→))∪(A←,A→)=(A←,A→)
(5)Distribution law
((A←,A→)∪(B←,B→))∩(C←,C→)=((A←,A→)∩((C←,C→)∪((B←,B→)∩(C←,C→))((A←,A→)∩(B←,B→))∪(C←,C→)=(A←,A→)∪((C←,C→)∩(B←,B→))∪(C←,C→))
(6)Zero-one law
(A←,A→)∪(ϕ←,ϕ→)=(A←,A→)(A←,A→)∩(ϕ←,ϕ→)=(ϕ←,ϕ→)(A←,A→)∪(U←,U→)=(U←,U→)(A←,A→)∩(U←,U→)=(A←,A→)
(7)Pull back law
((A←,A→)c)c=(A←,A→)