By Hung T. Nguyen

ISBN-10: 1584885262

ISBN-13: 9781584885269

A primary path in Fuzzy good judgment, 3rd version keeps to supply definitely the right advent to the idea and functions of fuzzy good judgment. This best-selling textual content offers an organization mathematical foundation for the calculus of fuzzy ideas useful for designing clever structures and an outstanding heritage for readers to pursue additional reviews and real-world functions.

New within the 3rd Edition:

With its entire updates, this new version offers the entire historical past invaluable for college students and pros to start utilizing fuzzy good judgment in its many-and speedily becoming- purposes in desktop technology, arithmetic, records, and engineering.

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

**Example text**

Then (u, v) and (v, w) belongTto each (v, w) ∈ i∈I Ei . Ei and T T hence (u, w) belongs to each Ei . Therefore, Thus i∈I Ei is a transitive relation on U . That i∈I Ei is reflexive and symmetric is similar. What we have shown is that the intersection of any family of equivalence relations on a set is an equivalence relation on that W set. This is clearly the inf of that family. Now {Ei : i ∈ I} of a family of equivalence relations on U is \ {E ∈ E(U ) : E ⊇ Ei for all i ∈ I} Note that U ×U is an equivalence containing all the Ei .

A(n) is a fuzzy subset of U , and trivially A(1) × ... × A(n) α = W (1) (n) Aα × ... × Aα . The fuzzy subset (A(1) × ... , A(n) ). , A(n) )α for all α > 0 if and only if for each member P of of the partition induced by f , _ (A(1) × ... × A(n) )(P ) = (A(1) × ... × A(n) )(u) 36 CHAPTER 2. SOME ALGEBRA OF FUZZY SETS for some u ∈ P . , un ) ∈ f (v) i=1 is attained. This result is often referred to as Nguyen’s Theorem in the literature. It is valid for L-fuzzy sets. Note that the extension of f : U (1) × ...

2 Let ∼ be an equivalence relation on a set U and let a ∈ U . The equivalence class of an element a is the set [a] = {u ∈ U : u ∼ a}. We defined a finite partition in Chapter 1. Here is the definition in general. 3 Let U be a nonempty set. A partition of U is a set of nonempty pairwise disjoint subsets of U whose union is U. There is an intimate connection between equivalence relations and partitions. Here are some examples illustrating these notions and this connection. 4 Let U be a set, and define x ∼ y if x = y.

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

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