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It was another great philosopher, Plato, who laid the foundations of what is today referred to as fuzzy logic It was, however, only in the 1900s that Lejewski and Lukasiewicz [514] proposed the rst alternative to the Aristotelian two-valued logic Three-valued logic has a third value which is assigned a numeric value between true and false Lukasiewicz later extended this to four-valued and ve-valued logic It was only recently, in 1965, that Lot Zadeh [944] produced the foundations of in nite-valued logic with his mathematics of fuzzy set theory Following the work of Zadeh, much research has been done in the theory of fuzzy systems, with applications in control, information systems, pattern recognition and decision support Some successful real-world applications include automatic control of dam gates for hydroelectric-powerplants, camera aiming, compensation against vibrations in camcorders, cruise-control for automobiles, controlling air-conditioning systems, document archiving systems, optimized planning of bus time-tables, and many more While fuzzy sets and logic have been used to solve real-world problems, they were also combined with other CI paradigms to form hybrid systems, for example, fuzzy neural networks and fuzzy genetic algorithms [957] A di erent set theoretic approach which also uses the concept of membership functions, namely rough sets (introduced by Pawlak in 1982 [668]), is sometimes confused with fuzzy sets While both fuzzy sets and rough sets make use of membership functions, rough sets di er in the sense that a lower and upper approximation to the rough set is determined The lower approximation consists of all elements that belong with full certainty to the corresponding set, while the upper approximation consists of elements that may possibly belong to the set Rough sets are frequently used in machine learning as classi er, where they are used to nd the smallest number of features to discern between classes [600] Rough sets are also used for extracting knowledge from incomplete data [600, 683] Hybrid approaches that employ both fuzzy and rough sets have also been developed [843] The remainder of this Part is organized as follows: 20 discusses fuzzy sets, while fuzzy logic and reasoning are covered in 21 A short overview of fuzzy controllers is given in 22 The Part is concluded with an overview of rough set theory in 23.

By adding (e) and (f),

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the message (for example, POP3 email works in this way and the Symbian OS smartphone requests messages from the email server). The SMS Client-side MTM class CSmsClientMtm uses the CBaseMtm::SetCurrentEntryL()andCBaseMtm::SwitchCurrentEntryL() methods to set the current context and then provides LoadMessageL() and SaveMessageL() methods to load or save a message. It also provides SmsHeader() methods to enable direct access to the SMS message header.

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Consider the problem of designing a set of all tall people, and assigning all the people you know to this set Consider classical set theory where an element is either a member of the set or not Suppose all tall people are described as those with height greater than 175m Then, clearly a person of height 178m will be an element of the set tall, and someone with height 15m will not belong to the set of tall people But, the same will apply to someone of height 173m, which implies that someone who falls only 2cm short is not considered as being tall Also, using two-valued set theory, there is no distinction among members of the set of tall people For example, someone of height 178m and one of height 21m belongs equally to the set! Thus, no semantics are included in the description of membership The alternative, fuzzy sets, has no problem with this situation In this case all the people you know will be members of the set tall, but to di erent degrees For example, a person of height 21m may be a member of the set to degree 095, while someone of length 17m may belong to the set with degree 04 Fuzzy sets are an extension of crisp (two-valued) sets to handle the concept of partial truth, which enables the modeling of the uncertainties of natural language The vagueness in natural language is further emphasized by linguistic terms used to describe objects or situations For example, the phrase when it is very cloudy, it will most probably rain, has the linguistic terms very and most probably which are understood by the human brain Fuzzy sets, together with fuzzy reasoning systems, give the tools to also write software, which enables computing systems to understand such vague terms, and to reason with these terms This chapter formally introduces fuzzy sets Section 201 de nes fuzzy sets, while membership functions are discussed in Section 202 Operators that can be applied to fuzzy sets are covered in Section 203 Characteristics of fuzzy sets are summarized in Section 204 The chapter is concluded with a discussion of the di erences between fuzziness and probability in Section 205.

By subtracting (f) from (e),

Computational Intelligence: An Introduction, Second Edition AP Engelbrecht c 2007 John Wiley & Sons, Ltd

Not all members and methods are described in this section: only those that are relevant to the work in this book are covered. If you want complete information then refer to the SDK or the relevant header les. One aspect that I do not cover in this chapter is asynchronous operations. The Messaging subsystem has some sophisticated APIs to support asynchronous operations, but I have opted to use the simpler synchronous methods. Actually, I have to use one asynchronous operation to send an SMS message, because there are no synchronous methods available. More detailed information on the Messaging APIs, including the extra methods for the classes covered here and the classes that are not covered here, can be found in SDKs.

Di erent to classical sets, elements of a fuzzy set have membership degrees to that set The degree of membership to a fuzzy set indicates the certainty (or uncertainty) that the element belongs to that set Formally de ned, suppose X is the domain, or universe of discourse, and x X is a speci c element of the domain X Then, the fuzzy set A is characterized by a membership mapping function [944] A : X [0, 1] (201)

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Therefore, for all x X, A (x) indicates the certainty to which element x belongs to fuzzy set A For two-valued sets, A (x) is either 0 or 1 Fuzzy sets can be de ned for discrete ( nite) or continuous (in nite) domains The notation used to denote fuzzy sets di er based on the type of domain over which that set is de ned In the case of a discrete domain X, the fuzzy set can either be expressed in the form of an nx -dimensional vector or using the sum notation If X = {x1 , x2 , , xnx }, then, using set notation, A = {( A (xi )/xi )|xi X, i = 1, , nx } Using sum notation,

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