# Extensions1 Extensions Definitions of fuzzy sets Definitions of fuzzy sets Operations with fuzzy sets Operations with fuzzy sets

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• Slide 1
• Extensions1 Extensions Definitions of fuzzy sets Definitions of fuzzy sets Operations with fuzzy sets Operations with fuzzy sets
• Slide 2
• Extensions2 Types of fuzzy sets Interval-valued fuzzy set Interval-valued fuzzy set Type two fuzzy set Type two fuzzy set Type m fuzzy set Type m fuzzy set L-fuzzy set L-fuzzy set
• Slide 3
• Extensions3 Interval-value fuzzy set 1 A membership function based on the latter approach does not assign to each element of the universal set one real number, but a closed interval of real numbers between the identified lower and upper bounds. A membership function based on the latter approach does not assign to each element of the universal set one real number, but a closed interval of real numbers between the identified lower and upper bounds. ([0,1]) closed intervals of real numbers in [0,1]. where ([0,1]) denotes the family of all closed intervals of real numbers in [0,1].
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• Extensions4 Interval-value fuzzy set 2
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• Extensions5 Type 2 fuzzy set 1 A fuzzy set whose membership values are type 1 fuzzy set on [0,1]. (fuzzy set whose membership function itself is a fuzzy set) A fuzzy set whose membership values are type 1 fuzzy set on [0,1]. (fuzzy set whose membership function itself is a fuzzy set)
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• Extensions6 Type 2 fuzzy set 2
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• Extensions7 Type m fuzzy set A fuzzy set in X whose membership values are type m-1 (m>1) fuzzy sets on [0,1]. A fuzzy set in X whose membership values are type m-1 (m>1) fuzzy sets on [0,1].
• Slide 8
• Extensions8 L-fuzzy set The membership function of an L-fuzzy set maps into a partially ordered set, L. The membership function of an L-fuzzy set maps into a partially ordered set, L.
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• Extensions9 A B A B R A B (a,b) R a,b R aRb A B A B R A B (a,b) R a,b R aRb R A R A a A aRa a A aRa aRb bRa aRb bRa aRb bRa a=b aRb bRa a=b aRb bRc aRc aRb bRc aRc
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• Extensions10 (partially ordered)
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• Extensions11 Lattice ( ) L (L ) L (L )
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• Extensions12 1 (L ) L a b=sup{a b} a b=inf{a b} (L ) L a b=sup{a b} a b=inf{a b} a a=a a a=a a a=a a a=a
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• Extensions13 2 a b=b a a b=b a a b=b a a b=b a (a b) c=a (b c) (a b) c=a (b c) (a b) c=a (b c) (a b) c=a (b c) a (a b)=a a (a b)=a a (a b)=a a (a b)=a
• Slide 14
• Extensions14 Operations on fuzzy sets Fuzzy complement Fuzzy complement Fuzzy intersection (t-norms) Fuzzy intersection (t-norms) Fuzzy union (t-conorms) Fuzzy union (t-conorms) Aggregation operations Aggregation operations
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• Extensions15 Fuzzy complements 1 C:[0,1][0,1] To produce meaningful fuzzy complements, function c must satisfy at least the following two axiomatic requirements: (axiomatic skeleton for fuzzy complements) Axiom c1. c(0)=1 and c(1)=0 (boundary conditions)( ) Axiom c2. For all a,b [0.1], if ab, then c(a)c(b) (monotonicity)( )
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• Extensions16 Fuzzy complements 2 Two of the most desirable requirements: Axiom c3. c is a continuous function.( ) Axiom c4. c is involutive, which means that c(c(a))=a for each a [0,1]( )
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• Extensions17 Fuzzy complements 3 Example Satisfy c1, c2 Satisfy c1,c2,c3 Satisfy c1~c4
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• Extensions18 Equilibrium of a fuzzy complement c Any value a for which c(a)=a Any value a for which c(a)=a Theorem 1: Every fuzzy complement has at most one equilibrium Theorem 1: Every fuzzy complement has at most one equilibrium Theorem 2: Assume that a given fuzzy complement c has an equilibrium e c, which by theorem 1 is unique. Then ac(a) iff aac(a) iff a Theorem 2: Assume that a given fuzzy complement c has an equilibrium e c, which by theorem 1 is unique. Then ac(a) iff ae c and ac(a) iff ae c Theorem 3: If c is a continuous fuzzy complement, then c has a unique equilibrium. Theorem 3: If c is a continuous fuzzy complement, then c has a unique equilibrium.
• Slide 19
• Extensions19 Fuzzy union (t-conorms) / Intersection (t-norms) Union u:[0,1] [0,1][0,1] Union u:[0,1] [0,1][0,1] Intersection i:[0,1] [0,1][0,1] Intersection i:[0,1] [0,1][0,1]
• Slide 20
• Extensions20 Axiomatic skeleton (t-conorms / t-norms) 1 Axiom u1/i1 (Boundary conditions) Axiom u1/i1 (Boundary conditions) u(0,0)=0; u(0,1)=u(1,0)=u(1,1)=1 u(0,0)=0; u(0,1)=u(1,0)=u(1,1)=1 i(1,1)=1; i(0,1)=i(1,0)=i(0,0)=0 i(1,1)=1; i(0,1)=i(1,0)=i(0,0)=0 Axiom u2/i2 (Commutative) Axiom u2/i2 (Commutative) u(a,b)=u(b,a) u(a,b)=u(b,a) i(a,b)=i(b,a) i(a,b)=i(b,a)
• Slide 21
• Extensions21 Axiomatic skeleton (t-conorms / t-norms) 2 Axiom u3/i3 (monotonic) If a Axiom u3/i3 (monotonic) If aa, bb U(a,b) U(a,b) u(a,b) i(a,b) i(a,b) Axiom u4/i4 (associative) Axiom u4/i4 (associative) u(u(a,b),c)=u(a,u(b,c)) u(u(a,b),c)=u(a,u(b,c)) i(i(a,b),c)=i(a,i(b,c)) i(i(a,b),c)=i(a,i(b,c))
• Slide 22
• Extensions22 Additional requirements (t-conorms / t-norms) 1 Axiom u5/i5 (continuous) Axiom u5/i5 (continuous) u/i is a continuous function u/i is a continuous function Axiom u6/i6 (idempotent)( ) Axiom u6/i6 (idempotent)( ) u(a,a)=i(a,a)=a u(a,a)=i(a,a)=a
• Slide 23
• Extensions23 Additional requirements (t-conorms / t-norms) 2 Axiom u7/i7 (subidempotency) Axiom u7/i7 (subidempotency) u(a,a)>a u(a,a)>a i(a,a)

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