Mathematics of Fuzzy Sets and Fuzzy Logic by Barnabas Bede

By Barnabas Bede

This publication offers a mathematically-based creation into the attention-grabbing subject of Fuzzy units and Fuzzy good judgment and can be used as textbook at either undergraduate and graduate degrees and likewise as reference consultant for mathematician, scientists or engineers who want to get an perception into Fuzzy good judgment.   Fuzzy units were brought through Lotfi Zadeh in 1965 and because then, they've been utilized in many functions. for this reason, there's a monstrous literature at the functional purposes of fuzzy units, whereas conception has a extra modest assurance. the most function of the current ebook is to minimize this hole via delivering a theoretical creation into Fuzzy units in accordance with Mathematical research and Approximation concept. famous functions, as for instance fuzzy keep watch over, also are mentioned during this publication and put on new flooring, a theoretical beginning. furthermore, a couple of complicated chapters and a number of other new effects are incorporated. those include, between others, a brand new systematic and confident technique for fuzzy inference structures of Mamdani and Takagi-Sugeno kinds, that investigates their approximation power through supplying new mistakes estimates.  

Show description

Read or Download Mathematics of Fuzzy Sets and Fuzzy Logic PDF

Similar microprocessors & system design books

Designing Embedded Systems with PIC Microcontrollers: Principles and Applications

This booklet is a hands-on creation to the foundations and perform of embedded method layout utilizing the PIC microcontroller. jam-packed with valuable examples and illustrations, it provides an in-depth remedy of microcontroller layout, programming in either meeting language and C, and contours complex themes corresponding to networking and real-time working platforms.

Logic and Language Models for Computer Science

This article makes in-depth explorations of a extensive diversity of theoretical themes in desktop technological know-how. It plunges into the purposes of the summary innovations as a way to confront and deal with the skepticism of readers, and instill in them an appreciation for the usefulness of idea. A two-part presentation integrates common sense and formal language—both with purposes.

Extra resources for Mathematics of Fuzzy Sets and Fuzzy Logic

Sample text

30. (see Acz´el [1]) The only continuous solution F : R → R of Cauchy’s functional equation F (x + y) = F (x) + F (y) is F (x) = cx, for some constant c ∈ R. Proof. We have F (0) = 2F (0), and so F (0) = 0. Then, F (n) = nF (1). Let m F (1) = c. , F (r) = cr, ∀r ∈ Q. Then we extend F by continuity and we get F (x) = cx, for any x ∈ R. 31. g. , there is a continuous strictly decreasing t : [0, 1] → [0, ∞], t(1) = 0, which is uniquely determined up to a multiplicative constant such that for all x, y ∈ [0, 1] we have T (x, y) = t(−1) (t(x) + t(y)).

Proof. (i) Case 1. If x ∨ y ≤ z then (x ∨ y) → z = 1 and also we have x ≤ z and y ≤ z. Then (x → z) ∧ (y → z) = 1 ∧ 1 = 1. Case 2. If x ∨ y > z then (x ∨ y) → z = z and x ≥ z or ,y ≥ z so we have (x → z) = z or (y → z) = z and then (x → z) ∧ (y → z) = z. (ii) Case 1. If x ∧ y ≤ z then (x ∧ y) → z = 1. Also, we have either x ≤ z or y ≤ z in which case either x → z = 1 or y → z = 1 and so (x → z) ∨ (y → z) = 1. Case 2. If x ∧ y > z, then (x ∧ y) → z = z. Also, if (x ∧ y) > z then both x > z and y > z and x → z = z and y → z = z.

23. The following inequalities hold true: (i) P ≤ R−1 (R ◦ P ); (ii) R ◦ (R−1 Q) ≤ Q; (iii) R ≤ (P (R ◦ P )−1 )−1 ; (iv) (P Q−1 )−1 ◦ P ≤ Q. Proof. 18, (vi) we have for every y ∈ Y and z ∈ Z, R−1 (R ◦ P )(y, z) = R−1 (y, x) → (R ◦ P )(x, z) x∈X R−1 (y, x) → = x∈X R(x, t) ∧ P (t, z) t∈Y R(x, y) → R(x, y) ∧ P (y, z) ≥ ≥ x∈X P (y, z) = P (y, z). 18, (v) we obtain for every x ∈ X and z ∈ Z R ◦ (R−1 Q)(x, z) = R(x, y) ∧ (R−1 Q)(y, z) y∈Y R(x, y) ∧ ( = y∈Y R−1 (y, t) → Q(t, z)) t∈X R(x, y) ∧ (R(x, y) → Q(x, z)) ≤ ≤ y∈Y Q(x, z) = Q(x, z).

Download PDF sample

Rated 4.68 of 5 – based on 22 votes