# Fuzzy Logic: An Introductory Course for Engineering Students (Studies in Fuzziness and Soft Computing)

## Enric Trillas

Language: English

Pages: 204

ISBN: 331914202X

Format: PDF / Kindle (mobi) / ePub

This book introduces readers to fundamental concepts in fuzzy logic. It describes the necessary theoretical background and a number of basic mathematical models. Moreover, it makes them familiar with fuzzy control, an important topic in the engineering field. The book offers an unconventional introductory textbook on fuzzy logic, presenting theory together with examples and not always following the typical mathematical style of theorem-corollaries. Primarily intended to support engineers during their university studies, and to spark their curiosity about fuzzy logic and its applications, the book is also suitable for self-study, providing a valuable resource for engineers and professionals who deal with imprecision and non-random uncertainty in real-world applications.

strong negation N , S is a t-conorm if and only if N ◦ S ◦ (N × N ) is a t-norm. Theorem 2.2.35 S is a continuous t-conorm if and only if, 1. 2. 3. 4. S = max, S is in the family of max S = pr odϕ∗ , S is in the family of prod∗ S = Wϕ∗ , S is in the family of W ∗ There exist an index set (finite or countable infinite), a family of pairwise disjoint open intervals of [0, 1], {(a1 , b1 ); i ∈ I }, and a family of t-conorms Si ∈ { pr od ∗ } ∪ {W ∗ }(i ∈ I ), such that S(x, y) = i , y−ai ), ai +

conditional a → b, through a · (a → b) b. Backwards reasoning goes from the consequent to the antecedent (also thanks to a → b), ad it is modeled by the Modus Tollens scheme. If a, then b not b not a, that is translated by b · (a → b) a ⇔a →b (b ) + a = a + b. Thus, in Boolean algebras, a → b = a + b, also allows backwards reasoning, provided b · (a → b) = b · (a + b) = a · b = 0, or a + b = 1. Nevertheless, although b · (a · b) = 0 a , it is clear that the conjunctive conditional a → b = a · b

T-transitive, and T1 is a t-norm such that T1 T , then μ R is also T1 -transitive, since T1 (μ R (x, y), μ R (y, z)) T (μ R (x, y), μ R (y, z)) μ R (x, z), for all x, y, z. 4.4 The Concept of T-State Given a fuzzy relation μ : X × X → [0, 1] and a continuous t-norm T , a fuzzy set σ : X → [0, 1] is a T -state of μ, if T (σ(x), μ(x, y)) σ(y), ∀(x, y) ∈ X × X. All constant fuzzy sets μk (x) = k for all x ∈ X , and k ∈ [0, 1], are T-states of any fuzzy relation μ : X × X → [0, 1]: T (μk (x),

min(μ(x), μ B (x))) x∈X = max(Sup min(μ(x), μ A (x)), Sup min(μ(x), μ B (x))) x∈X x∈X = max(πμ (A), πμ (B)). It is said that μ is the possibility distribution of πμ , a possibility measure that can be considered as the one “conditioned” by μ. Thus, each fuzzy set μ ∈ [0, 1] X such that Sup μ = 1 (μ is never self-contradictory) induces the possibility measure πμ . Provided X = {x1 , . . . , xn } is a finite set, it results πμ (A) = Max min(μ(xi ), μ A (xi )). 1 i n Hence, all non

Mamdani-Larsen implications do verify μ Q ∗ = μ Q whenever μ P ∗ = μ P . Fourth Step: Numerical Input This is the case in which μ P ∗ is exactly x = x0 or x ∈ {x0 }. That is “x is P ∗ ” is the statement “x is x0 ” and hence μ P ∗ (x) = 1 0 if x = x0 if x = x0 In that case, μ Q ∗ (y) = SupT1 (μ P j∗ (x), J (μ P (x), μ Q (y))) = J (μ P (x0 ), μ Q (y)), x∈X for all y ∈ Y . For example, let J be a ML-implication, then μ Q ∗ (y) = T (μ P (x0 ), μ Q (y)). If X = [0, 10], Y = [0, 1], P = close to 4,