By Professor James J. Buckley, Professor Esfandiar Eslami (auth.)

This publication is to be the place to begin for any curriculum in fuzzy platforms in fields like computing device technology, arithmetic, business/economics and engineering. It covers the fundamentals resulting in: fuzzy clustering, fuzzy trend acceptance, fuzzy database, fuzzy snapshot processing, smooth computing, fuzzy purposes in operations examine, fuzzy selection making, fuzzy rule dependent platforms, fuzzy structures modeling, fuzzy arithmetic. it isn't a publication designed for researchers - it's the place you actually study the "basics" wanted for any of the above-mentioned applications.

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**Extra resources for An Introduction to Fuzzy Logic and Fuzzy Sets**

**Example text**

Determine if An (AU B) = A holds if we use: a. Tm and Cm. b. Tp and Cp. 4. ALGEBRA OF FUZZY SETS 41 c. Tb and Cb. d. T* and C*. 10. 17)) that must hold for fuzzy sets when we use: a. Tm and Cm. b. Tp and Cp. c. Tb and Cb. d. T* and C*. CHAPTER 3. 5 Mixed Fuzzy Logic Most people use T m and Cm in all their calculations with fuzzy sets. However, you need not always use the same t-norm and t-conorm. Mixed fuzzy logic is when you use T = T1 , C = C 1 , usually dual operators, for certain calculations and switch to T = T2 , C = C 2 , probably also dual operators, for other calculations.

We start with Tb(a1,az,a3) = n(a1,T(az,a3)) which is max(O, a1 + max(O, az + a3 - 1) - 1). 55) for n = 3. 58) + a3 - 1 < 0. 3. T -NORMS, T -CONORMS 35 which is zero since a 1 :S: 1. 55), for n = 3, is also zero since a1 + a2 + a3 - 2 < 0 when a2 + a3 - 1 < 0. Next we look at Cb(a1,a2,a3) = Cb(a1,Cb(a2,a3)) which equals min(1, a1 + min(1, a2 + a3)). 61). So let a 2 + a 3 > 1. 61) equal one. 62) but the formula for Cp is more complicated. When n = 3 we see CHAPTER 3. FUZZY SETS 36 3. 1 Exercises 1.

Now consider S' = {xiO ::; x < 1} which has no maximum member so that max S' does not exist. In this case we would use sup and supS'= 1. The least upper bound of S' is called supS'. A number u is called an upper bound for S' if x ::; u for all x in S' and supS' is the smallest upper bound. Any subset of R that has an upper bound has a supremum. Therefore, "sup" is more general than max and supS = maxS = 1. So, in general, we will use "sup" even when 24 CHAPTER 3. FUZZY SETS the set can have a maximum value.