Download e-book for kindle: A First Course in Fuzzy Logic, Third Edition by Hung T. Nguyen

By Hung T. Nguyen

ISBN-10: 1584885262

ISBN-13: 9781584885269

A primary path in Fuzzy common sense, 3rd variation keeps to supply the correct creation to the idea and functions of fuzzy good judgment. This best-selling textual content offers a company mathematical foundation for the calculus of fuzzy ideas invaluable for designing clever platforms and a superb historical past for readers to pursue extra reports and real-world functions.

New within the 3rd Edition:

  • A part on type-2 fuzzy units - an issue that has acquired a lot realization long ago few years
  • Additional fabric on copulas and t-norms
  • More discussions on generalized modus ponens and the compositional rule of inference
  • Complete revision to the bankruptcy on hazard theory
  • Significant growth of the bankruptcy on fuzzy integrals
  • Many new exercises

    With its accomplished updates, this new version offers the entire historical past important for college students and pros to start utilizing fuzzy common sense in its many-and quickly becoming- functions in computing device technological know-how, arithmetic, data, and engineering.
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    Additional resources for A First Course in Fuzzy Logic, Third Edition

    Example text

    7. EXERCISES 43 i. For each w ∈ W, sup{T (A(u), B(v)) : (u, v) ∈ f −1 (w)} is attained. S ii. For each α ∈ (, 1], [fT (A, B)]α = T (t,s)≥α f (At , Bs ). (b) If T (x, y) = x ∧ y, verify that [fT (A, B)]α = f (Aα , Bα ) . S (c) If T (x, y) = xy, verify that [fT (A, B)]α = t∈[α,1] f (At , Bα/t ). (d) If T (x, y) = x if y = 1, y if x = 1, and 0 otherwise, verify that [fT (A, B)]α = f (A1 , Bα ) ∪ f (Aα , B1 ) (e) If T (x, y) = max{0, x + y − 1}, verify that [ [fT (A, B)]α = f (At , Bα+1−t ) t∈[α,1] 46.

    So knowing all the α-cuts of A is the same as knowing A itself. We can state this as follows. 2 Let A and B be mappings from a set U into a partially ordered set C. If Aα = Bα for all α ∈ C, then A = B. 5. ALPHA-CUTS 33 Thus we have the following situation. A function A : U → C gets a function A−1 ↑ : C → P(U ) which is just the composition A−1 ↑ C −→ P(C) −→ P(U ) We already know from the theorem just above that associating A with the function A−1 ↑ is one-to-one. What we want first is a simple description of the functions C → P(U ) that can come from functions A : U → C in this way.

    The equation x∗ ∨ x∗∗ = 1 is called Stone’s identity, and a Stone algebra is a pseudocomplemented distributive lattice satisfying this identity. If (S, ∨, ∧,∗ , 0, 1) is a Stone algebra, then for S ∗ = {s∗ ∈ S : s ∈ S}, (S ∗ , ∨, ∧,∗ , 0, 1) is a Boolean algebra. That is, ∗ is a complement on S ∗ . The sublattice S ∗ consists precisely of the complemented elements of S, and is sometimes called the center of S. If B is a Boolean algebra, then B [2] is a Stone algebra (Exercise 12). Stone algebras have a fairly extensive theory [64] and only a few facts are cited here and in the exercises.

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