By Hung T. Nguyen

ISBN-10: 1584885262

ISBN-13: 9781584885269

A primary direction in Fuzzy common sense, 3rd version keeps to supply the suitable creation to the speculation and functions of fuzzy good judgment. This best-selling textual content presents an organization mathematical foundation for the calculus of fuzzy techniques worthy for designing clever structures and a high-quality heritage for readers to pursue extra reviews and real-world purposes.

New within the 3rd Edition:

With its accomplished updates, this re-creation offers all of the heritage useful for college students and execs to start utilizing fuzzy good judgment in its many-and speedily transforming into- functions in machine technological know-how, arithmetic, facts, and engineering.

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**Additional info for A First Course in Fuzzy Logic, Third Edition**

**Sample text**

A ∨ b) ∨ c = a ∨ (b ∨ c) and (a ∧ b) ∧ c = a ∧ (b ∧ c). ) 4. a ∨ (a ∧ b) = a and a ∧ (a ∨ b) = a. ) The proof is quite easy and is left as an exercise. A pertinent fact is that two binary operations satisfying conditions 1-4 define a lattice. 4, then defining a ≤ b if a ∧ b = a makes (U, ≤) into a lattice whose sup and inf operations are ∨ and ∧. Proof. We first show that a ∧ b = a if and only if a ∨ b = b. Thus defining a ≤ b if a ∧ b = a is equivalent to defining a ≤ b if a ∨ b = b. Indeed, if a ∧ b = a, then a ∨ b = (a ∧ b) ∨ b = b by one of the absorption laws.

Let N be the set of positive integers and let R be the relation mRn if m divides n. Show that this makes N into a distributive lattice. 8. 6 9. Show that the De Morgan algebra (F(U ), ∨, ∧,0 , 0, 1) satisfies A ∧ A0 ≤ B ∨ B 0 for all A, B ∈ F(U ), that is, is a Kleene algebra. Show that [0, 1] is a Kleene algebra. Show that [0, 1][2] is not a Kleene algebra. 10. Show that the product X × Y of lattices X and Y is a lattice. Show that X × Y is respectively, bounded, complete, distributive, complemented, Boolean, or De Morgan if and only if X and Y are.

2 If ∼ is a congruence on the lattice U, then the set of equivalence classes U/ ∼ forms a lattice under the operations [a] ∨ [b] = [a ∨ b] and [a] ∧ [b] = [a ∧ b]. The mapping U → U/ ∼: a → [a] is a lattice homomorphism. 4. ISOMORPHISMS AND HOMOMORPHISMS 31 The proof is left as an exercise. The lattice U/ ∼ is the quotient lattice of U relative to the congruence ∼ . Congruences are defined analogously on other algebraic systems, such as Boolean algebras, De Morgan algebras, Stone algebras, and so on.

### A First Course in Fuzzy Logic, Third Edition by Hung T. Nguyen

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