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## Extensions

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**Extensions**• Definitions of fuzzy sets • Operations with fuzzy sets Extensions**Types of fuzzy sets**• Interval-valued fuzzy set • Type two fuzzy set • Type m fuzzy set • L-fuzzy set Extensions**Interval-value fuzzy set1**• 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. where ([0,1]) denotes the family of all closed intervals of real numbers in [0,1]. Extensions**Interval-value fuzzy set2**Extensions**Type 2 fuzzy set1**• A fuzzy set whose membership values are type 1 fuzzy set on [0,1]. (fuzzy set whose membership function itself is a fuzzy set) Extensions**Type 2 fuzzy set2**Extensions**Type m fuzzy set**• A fuzzy set in X whose membership values are type m-1 (m>1) fuzzy sets on [0,1]. Extensions**L-fuzzy set**• The membership function of an L-fuzzy set maps into a partially ordered set, L. Extensions**集合之二元關係**• 給定集合A及集合B，直積AB的每個子集R都叫從A到B的關係，當(a,b)R時，稱a,b適合關係R，記作aRb。 • 設非空的R是集合A上的二元關係 • 每個aA都有aRa，則稱其具備自反性。 • 若由aRb必可推出bRa，則稱其具備對稱性。 • 若由aRb及bRa必可推出a=b，則稱其具備反對稱性。 • 若由aRb及bRc必可推出aRc，則稱其具備傳遞性。 Extensions**偏序(partially ordered)集**• 集合上具備自反性、反對稱性及傳遞性的關係叫做偏序關係。 • 具偏序關係之集合稱為偏序集。 Extensions**Lattice (格)**• 若非空之偏序集L的任何個元素構成的集合，既有上確界也有下確界，則偏序集(L，≤)叫做格。 Extensions**格的主要性質1**• 設(L，≤)是格如果在L上規定二元運算∨及∧，a∨b=sup{a，b} ，a∧b=inf{a，b} ，則此二元運算滿足如下算律。 • 冪等律 • a∨a=a • a∧a=a Extensions**格的主要性質2**• 交換律 • a∨b=b∨a • a∧b=b∧a • 結合律 • (a∨b)∨c=a∨(b∨c) • (a∧b) ∧c=a∧(b∧c) • 吸收律 • a∧(a∨b)=a • a∨(a∧b)=a Extensions**Operations on fuzzy sets**• Fuzzy complement • Fuzzy intersection (t-norms) • Fuzzy union (t-conorms) • Aggregation operations Extensions**Fuzzy complements1**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)(邊界條件) • Axiom c2. • For all a,b [0.1], if a≤b, then c(a)≥c(b) (monotonicity)(單調) Extensions**Fuzzy complements2**Two of the most desirable requirements: • Axiom c3. • c is a continuous function.(連續) • Axiom c4. • c is involutive, which means that c(c(a))=a for each a [0,1](復原) Extensions**Satisfy c1, c2**Satisfy c1,c2,c3 Satisfy c1~c4 Fuzzy complements3Example Extensions**Equilibrium of a fuzzy complement c**• Any value a for which c(a)=a • Theorem 1: Every fuzzy complement has at most one equilibrium • Theorem 2: Assume that a given fuzzy complement c has an equilibrium ec, which by theorem 1 is unique. Then a≤c(a) iff a≤ec and a≥c(a) iff a≥ec • Theorem 3: If c is a continuous fuzzy complement, then c has a unique equilibrium. Extensions**Fuzzy union (t-conorms) / Intersection (t-norms)**• Union u:[0,1][0,1]→[0,1] • Intersection i:[0,1][0,1]→[0,1] Extensions**Axiomatic skeleton(t-conorms / t-norms)1**• Axiom u1/i1 (Boundary conditions) • 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 • Axiom u2/i2 (Commutative) • u(a,b)=u(b,a) • i(a,b)=i(b,a) Extensions**Axiomatic skeleton(t-conorms / t-norms)2**• Axiom u3/i3 (monotonic) If a≤a’, b≤b’ • U(a,b) ≤u(a’,b’) • i(a,b) ≤i(a’,b’) • Axiom u4/i4 (associative) • u(u(a,b),c)=u(a,u(b,c)) • i(i(a,b),c)=i(a,i(b,c)) Extensions**Additional requirements(t-conorms / t-norms)1**• Axiom u5/i5 (continuous) • u/i is a continuous function • Axiom u6/i6 (idempotent)(冪等) • u(a,a)=i(a,a)=a Extensions**Additional requirements(t-conorms / t-norms)2**• Axiom u7/i7 (subidempotency) • u(a,a)>a • i(a,a)<a • Axiom u8/i8 (strict monotonicity) • a1<a2 and b1<b2 implies u(a1,b1)<u(a2,b2) • a1<a2 and b1<b2 implies i(a1,b1)<i(a2,b2) • Theorem 4 • The standard fuzzy union/intersection is the onlyidempotent and continuous t-conorm/t-norm (i.e., the only function that satisfies Axiom u1/i1~u6/i6) Extensions**Some t-conorms/t-norms(Drastic union/intersection1)**Extensions**Cartesian Product of fuzzy set**• mth power of a fuzzy set Some t-conorms/t-norms2 Extensions**Algebraic sum**where Some t-conorms/t-norms3 Extensions**Bounded sum**where Some t-conorms/t-norms4 Extensions**Bounded difference**where Some t-conorms/t-norms5 Extensions**Algebraic product**where Some t-conorms/t-norms6 Extensions**Example**• Let Extensions**Some theorem of t-conorms/t-norms**• Theorem 5 • For all a,b [0,1], u(a,b) ≥max(a,b) • Theorem 6 • For all a,b [0,1], u(a,b) ≤umax(a,b) • Theorem 7 • For all a,b [0,1], i(a,b) ≤min(a,b) • Theorem 8 • For all a,b [0,1], i(a,b) ≥imin(a,b) Extensions**Aggregation operations**• Definition • Operations by which several fuzzy sets are combined to produce a single set. h:[0,1]n→[0,1] for n≥2 Extensions**Axiomatic requirements**• Axiom h1 • h(0,0,…,0)=0 and h(1,1,…,1)=1 (boundary conditions) • Axiom h2 • For any pair (ai,bi), ai,bi[0,1] if ai≥bi i, then h(ai) ≥h(bi) (monotonic increasing) • Axiom h3 • h is a continuous function. Extensions**Additional axiomatic requirements**• Axiom h4 • h is a symmetric function in all its arguments; that is, h(a1,a2,…,an)=h(ap(1),ap(2),…,ap(n)) for any permutation p on Nn. • Axiom h5 • H is an idempotent function; that is, h(a,a,…,a)=a Extensions**Averaging operations**• Any aggregation operation h that satisfies Axioms h2 and h5 satisfies also the inequalities: • min(a1,a2,…,an)≤h(a1,a2,…,an) ≤max(a1,a2,…,an) • All aggregations between the standard fuzzy intersection and the standard fuzzy union are idempotent. • These aggregation operations are usually called averaging operations. Extensions**Example for averaging operation**• Generalized means (satisfies h1 through h4) α:parameter, αR (α≠0) Extensions**Criteria for selecting appropriate aggregation operators**• Axiomatic strength • Empirical fit • Adaptability • Numerical efficiency • Compensation • Range of compensation • Aggregating behavior • Required scale level of membership functions Extensions