Artificial Intelligence

A First Course in Fuzzy Logic by Hung T. Nguyen, Elbert A. Walker

By Hung T. Nguyen, Elbert A. Walker

Utilizing fabric from a winning path on fuzzy good judgment, this e-book is an creation to the idea of fuzzy units: mathematical items modeling the vagueness of our common language once we describe phenomena that don't have sharply outlined barriers. The publication offers history info essential to observe fuzzy set idea in quite a few components, together with engineering, fuzzy common sense, and determination making. The routines on the finish of every bankruptcy serve to deepen the reader's knowing of the innovations, and to check their skill to make the required calculations.

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When are these partially ordered sets “just alike”? For example, it is intuitively clear that as partially ordered sets, there is no difference between ([0, 1], ≤) and ([1, 2], ≤). For them to be alike, there must be a one-to-one mapping from U onto V that respects, in some sense, the ordering of the two sets. Precisely, there must be a one-to-one onto mapping f : U → V such that x ≤ y if and only if f (x) ≤ f (y). Such a mapping f is an (order) isomorphism, and if there is such a mapping from U to V, the partially ordered sets (U, ≤) and (V, ≤) are isomorphic.

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. This example is just meant to point out that equality is an equivalence relation. For any x ∈ U , [x] = {x}. That is, the equivalence classes are just singletons. But do notice that the equivalence classes form a partition of U . 5 Let U be a set and define x ∼ y for any two elements of U . That is, any two elements are equivalent.

If A(x) + ≤ 1 then for α = A(x)+ , x ∈ / Aα and so there is δ > 0 such that (x−δ, x+δ)∩Aα = ∅. Thus A(y) < α = A(x) + for all y with |x − y| < δ. There is δ > 0 Conversely, take α ∈ [0, 1], x ∈ / Aα , and = α−A(x) 2 such that |x − y| < δ implies that A(y) < A(x) + α−A(x) < α, and so 2 (x − δ, x + δ) ∩ Aα = ∅. Thus Aα is closed. The following theorem is the crucial fact that enables us to use α-cuts in computing with fuzzy quantities. 9 Let ◦ : R×R → R be a continuous binary operation on R and let A and B be fuzzy quantities with closed α-cuts and bounded supports.

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