By Hung T. Nguyen, Elbert A. Walker
Utilizing fabric from a profitable direction on fuzzy good judgment, this booklet is an creation to the speculation of fuzzy units: mathematical gadgets modeling the vagueness of our ordinary language once we describe phenomena that don't have sharply outlined limitations. The publication offers historical past info essential to observe fuzzy set concept in numerous parts, 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 options, and to check their skill to make the required calculations.
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Additional resources for First Course in Fuzzy Logic
Example text
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. Similarly, if a ∨ b = b, then a ∧ b = a. We show the existence of sups, and claim that sup{a, b} = a ∨ b. Now a ≤ a ∨ b since a ∧ (a ∨ b) = a by one of the absorption laws. Similarly b ≤ b ∨ a = a ∨ b, so that a ∨ b is an upper bound of a and b. For any other upper bound x, a = a ∧ x and b = b∧x, whence x = a∨x = b∨x. Therefore, x = a∨x∨b∨x = (a∨b)∨x, and so a ∨ b ≤ x.
However, B [2] does have pseudocomplements. 2. 8 Let X be a bounded lattice, and let x ∈ X. Then an element x∗ is a pseudocomplement of x if x ∧ x∗ = 0, and y ≤ x∗ whenever x ∧ y = 0. That is, for each x ∈ X, there is a unique largest element whose meet with x is 0. An element in a bounded lattice has at most one pseudocomplement since two pseudocomplements must each be less or equal to the other, and hence equal. If every element has a pseudocomplement, then the bounded lattice is pseudocomplemented, and the unary operation ∗ is called a pseudocomplement.
In particular, a real number may be divided by 0 in F(R). Recall that R is viewed inside F(R) as the characteristic functions χ{r} for elements r of R. We note the following easy proposition. 1 For any fuzzy set A, A/χ{0} is the constant function whose value is A(0). 1. FUZZY QUANTITIES 47 Proof. The function A/χ{0} is given by the formula ´ ³ ´ W ³ A/χ{0} (u) = A(s) ∧ χ{0} (t) s=t·u ´ W ³ = A(s) ∧ χ{0} (0) s=0·u = A(0) Thus χ{r} /χ{0} is the constant function 0 if r 6= 0 and 1 if r = 0. Neither of these fuzzy quantities is real numbers, that is, neither is a characteristic function of a real number.