PART 3 Operations on fuzzy sets FUZZY SETS AND FUZZY LOGIC Theory and Applications PART 3 Operations on fuzzy sets 1. Fuzzy complements 2. Fuzzy intersections 3. Fuzzy unions 4. Combinations of operations 5. Aggregation operations

(boundary conditions). Fuzzy complements Axiomatic skeleton Axiom c1. Axiom c2. (boundary conditions). For all if , then (monotonicity).

Fuzzy complements Desirable requirements Axiom c3. Axiom c4. c is a continuous function. c is involutive, which means that for each

Fuzzy complements Theorem 3.1 Let a function satisfy Axioms c2 and c4. Then, c also satisfies Axioms c1 and c3. Moreover, c must be a bijective function.

Fuzzy complements

Fuzzy complements

Fuzzy complements Sugeno class Yager class

Fuzzy complements

Fuzzy complements Theorem 3.2 Every fuzzy complement has at most one equilibrium.

Fuzzy complements Theorem 3.3 Assume that a given fuzzy complement c has an equilibrium ec , which by Theorem 3.2 is unique. Then and

Fuzzy complements Theorem 3.4 Theorem 3.5 If c is a continuous fuzzy complement, then c has a unique equilibrium. If a complement c has an equilibrium ec , then

Fuzzy complements

Fuzzy complements Theorem 3.6 For each , , that is, when the complement is involutive.

Fuzzy complements Theorem 3.7 (First Characterization Theorem of Fuzzy Complements). Let c be a function from [0, 1] to [0, 1]. Then, c is a fuzzy complement (involutive) iff there exists a continuous function from [0, 1] to R such that , is strictly increasing, and for all

Fuzzy complements Increasing generators Sugeno: Yager:

Fuzzy complements Theorem 3.8 (Second Characterization Theorem of Fuzzy complements). Let c be a function from [0, 1] to [0, 1]. Then c is a fuzzy complement iff there exists a continuous function from [0, 1] to R such that , is strictly decreasing, and for all .

Fuzzy complements Decreasing generators Sugeno: Yager:

Fuzzy intersections: t-norms Axiomatic skeleton Axiom i1. Axiom i2. (boundary condition). implies (monotonicity).

Fuzzy intersections: t-norms Axiomatic skeleton Axiom i3. Axiom i4. (commutativity). (associativity).

Fuzzy intersections: t-norms Desirable requirements Axiom i5 Axiom i6 Axiom i7 is a continuous function (continuity). (subidempotency). implies (strict monotonicity).

Fuzzy intersections: t-norms Archimedean t-norm: A t-norm satisfies Axiom i5 and i6. Strict Archimedean t-norm: Archimedean t-norm and satisfies Axiom i7.

Fuzzy intersections: t-norms Frequently used t-norms

Fuzzy intersections: t-norms

Fuzzy intersections: t-norms

Fuzzy intersections: t-norms Theorem 3.9 Theorem 3.10 The standard fuzzy intersection is the only idempotent t-norm. For all , where denotes the drastic intersection.

Fuzzy intersections: t-norms Pseudo-inverse of decreasing generator The pseudo-inverse of a decreasing generator , denoted by , is a function from R to [0, 1] given by where is the ordinary inverse of .

Fuzzy intersections: t-norms Pseudo-inverse of increasing generator The pseudo-inverse of a increasing generator , denoted by , is a function from R to [0, 1] given by where is the ordinary inverse of .

Fuzzy intersections: t-norms Lemma 3.1 Let be a decreasing generator. Then a function defined by for any is an increasing generator with , and its pseudo-inverse is given by for any R.

Fuzzy intersections: t-norms Lemma 3.2 Let be a increasing generator. Then a function defined by for any is an decreasing generator with , and its pseudo-inverse is given by for any R.

Fuzzy intersections: t-norms Theorem 3.11 (Characterization Theorem of t-Norms). Let be a binary operation on the unit interval. Then, is an Archimedean t-norm iff there exists a decreasing generator such that for all .

Fuzzy intersections: t-norms [Schweizer and Sklar, 1963]

Fuzzy intersections: t-norms [Yager, 1980f]

Fuzzy intersections: t-norms [Frank, 1979]

Fuzzy intersections: t-norms Theorem 3.12 Let denote the class of Yager t-norms. Then, for all , where the lower and upper bounds are obtained for and ,respectively.

Fuzzy intersections: t-norms Theorem 3.13 Let be a t-norm and be a function such that is strictly increasing and continuous in (0, 1) and Then, the function defined by for all ,where denotes the pseudo-inverse of , is also a t-norm.

Fuzzy unions: t-conorms Axiomatic skeleton Axiom u1. Axiom u2.

Fuzzy unions: t-conorms Axiomatic skeleton Axiom u3. Axiom u4.

Fuzzy unions: t-conorms Desirable requirements Axiom u5. Axiom u6. Axiom u7.

Fuzzy unions: t-conorms Frequently used t-conorms

Fuzzy unions: t-conorms

Fuzzy unions: t-conorms

Fuzzy unions: t-conorms Theorem 3.14 The standard fuzzy union is the only idempotent t-conorm.

Fuzzy unions: t-conorms Theorem 3.15 For all

Fuzzy unions: t-conorms Theorem 3.16 (Characterization Theorem of t-Conorms). Let u be a binary operation on the unit interval. Then, u is an Archimedean t-conorm iff there exists an increasing generator such that for all

Fuzzy unions: t-conorms [Schweizer and Sklar, 1963]

Fuzzy unions: t-conorms [Yager, 1980f]

Fuzzy unions: t-conorms [Frank, 1979]

Fuzzy unions: t-conorms Theorem 3.17 Let uw denote the class of Yager t-conorms. for all where the lower and upper bounds are obtained for , respectively.

Fuzzy unions: t-conorms Theorem 3.18 Let u be a t-conorm and let be a function such that is strictly increaning and continuous in (0, 1) and . Then, the function defined by for all is also a t-conorm.

Combinations of operators Theorem 3.19 The triples 〈min, max, c〉and〈imin, umax, c〉are dual with respect to any fuzzy complement c.

Combinations of operators Theorem 3.20 Given a t-norm i and an involutive fuzzy complement c, the binary operation u on [0, 1] defined by for all is a t-conorm such that 〈i, u, c〉is a dual triple.

Combinations of operators Theorem 3.21 Given a t-conorm u and an involutive fuzzy complement c, the binary operation i on [0, 1] defined by for all is a t-norm such that 〈i, u, c〉is a dual triple.

Combinations of operators Theorem 3.22 Given an involutive fuzzy complement c and an increasing generator of c, the t-norm and t-conorm generated by are dual with respect to c.

Combinations of operators Theorem 3.23 Let〈i, u, c〉be a dual triple generated by Theorem 3.22. Then, the fuzzy operations i, u, c satisfy the law of excluded middle and the law of contradiction.

Combinations of operators Theorem 3.24 Let〈i, u, c〉be a dual triple that satisfies the law of excluded middle and the law of contradiction. Then,〈i, u, c〉does not satisfy the distributive laws.

Aggregation operations Axiomatic requirements Axiom h1. Axiom h2.

Aggregation operations Axiomatic requirements Axiom h3.

Aggregation operations Additional requirements Axiom h4. Axiom h5.

Aggregation operations Theorem 3.25

Aggregation operations Theorem 3.26

Aggregation operations Theorem 3.27

Aggregation operations Theorem 3.28

Exercise 3 3.6 3.7 3.13 3.14