By Barnabas Bede
ISBN-10: 3642352200
ISBN-13: 9783642352201
This ebook offers a mathematically-based creation into the interesting subject of Fuzzy units and Fuzzy good judgment and can be used as textbook at either undergraduate and graduate degrees and in addition as reference consultant for mathematician, scientists or engineers who wish to get an perception into Fuzzy common sense. Fuzzy units were brought via Lotfi Zadeh in 1965 and because then, they've been utilized in many functions. in this case, there's a titanic literature at the useful functions of fuzzy units, whereas idea has a extra modest insurance. the most objective of the current e-book is to minimize this hole by way of delivering a theoretical creation into Fuzzy units in line with Mathematical research and Approximation thought. famous purposes, as for instance fuzzy keep an eye on, also are mentioned during this publication and put on new floor, a theoretical origin. in addition, a couple of complex chapters and several other new effects are integrated. those contain, between others, a brand new systematic and positive procedure for fuzzy inference structures of Mamdani and Takagi-Sugeno kinds, that investigates their approximation strength by way of offering new errors estimates.
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Extra info for Mathematics of Fuzzy Sets and Fuzzy Logic
Sample text
2. The fuzzy set u : R → [0, 1], ⎧ 0 if x<0 ⎪ ⎪ ⎨ x3 if 0 ≤ x < 1 u(x) = (2 − x)3 if 1 ≤ x ≤ 2 ⎪ ⎪ ⎩ 0 if x≥2 is a fuzzy number. 1. B. Bede: Mathematics of Fuzzy Sets and Fuzzy Logic, STUDFUZZ 295, pp. 51–64. 1007/978-3-642-35221-8_4 52 4 Fuzzy Numbers Fig. 1 Example of a fuzzy number and its level sets Fig. 3. 2 is a fuzzy number as well. 4. 3 is not a fuzzy number since it is not fuzzy convex. 1 Definition of Fuzzy Numbers 53 Fig. 3 Example of a fuzzy set that is not a fuzzy number Fig. 5. , R ⊂ RF .
25. 5 ⎠ . 6 then R−1 Q is a solution of the equation. From the previous theorem it follows that it is the greatest solution of the given equation. 26. The following inequalities hold true: (i) (Q P −1 ) P ≥ Q; (ii) (R P ) P −1 ≥ R; (iii) R−1 ◦ (R P ) ≤ P ; (iv) R (R−1 ◦ Q) ≥ Q. Proof. 18, (vii), and taking into account that → is decreasing in its first argument we have (Q P −1 ) P (x, z) = (Q P −1 )(x, y) → P (y, z) y∈Y = Q(x, t) → P (y, t)) → P (y, z) ( y∈Y t∈Z (Q(x, z) → P (y, z)) → P (y, z) ≥ ≥ y∈Y Q(x, z) = Q(x, z).
Then (2n) (n) (n) xT = T (xT , xT ) and since T is continuous we have (2n) lim xT n→∞ (n) (n) (n) (n) = lim T (xT , xT ) = T ( lim xT , lim xT ). n→∞ n→∞ n→∞ (2n) is also y, we get y = T (y, y). But Taking into account that limn→∞ xT from the hypothesis, T (y, y) < y, so we get a contradiction. (ii) The proof of (ii) is similar. 25. The product t-norm TG (x, y) = x · y is Archimedean, while the min t-norm TM (x, y) = x ∧ y is not Archimedean. 26. (Klement-Mesiar-Pap [89]) Let f : [a, b] → [c, d] be monotonic, and [a, b], [c, d] ⊂ [−∞, ∞].
Mathematics of Fuzzy Sets and Fuzzy Logic by Barnabas Bede
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