FUZZY LOGIC & FUZZY SETS

1 FUZZY LOGIC & FUZZY SETSSiti Zaiton Mohd Hashim, PhD2 FUZZY logicFuzzy LOGIC Introduction: Introduction: what is FUZZY thinking?what is FUZZY thinking? FUZZY setsFuzzy sets Linguistic variables and hedgesLinguistic variables and hedges Operations of FUZZY setsOperations of FUZZY sets FUZZY rulesFuzzy rules SummarySummary3 What is FUZZY thinking?What is FUZZY thinking? Experts rely on Experts rely on common sensecommon sensewhen they solve when they solve problems. problems. How can we represent expert knowledge that How can we represent expert knowledge that uses vague and ambiguous terms in a computer?uses vague and ambiguous terms in a computer? FUZZY LOGIC is not LOGIC that is FUZZY , but LOGIC that FUZZY LOGIC is not LOGIC that is FUZZY , but LOGIC that is used to describe fuzziness.

2 FUZZY LOGIC is the is used to describe fuzziness. FUZZY LOGIC is the theory of FUZZY sets, sets that calibrate of FUZZY sets, sets that calibrate vagueness. FUZZY LOGIC is based on the idea that all things FUZZY LOGIC is based on the idea that all things admit of degrees. Temperature, height, speed, admit of degrees. Temperature, height, speed, distance, beauty distance, beauty all come on a sliding scale. The all come on a sliding scale. The motor is running motor is running really hotreally hot. Tom is a . Tom is a very tallvery Boolean LOGIC uses sharp distinctions. It forces us Boolean LOGIC uses sharp distinctions. It forces us to draw lines between members of a class and nonto draw lines between members of a class and non--members.

3 For instance, we may say, Tom is tall members. For instance, we may say, Tom is tall because his height is 181 cm. If we drew a line at because his height is 181 cm. If we drew a line at 180 cm, we would find that David, who is 179 cm, 180 cm, we would find that David, who is 179 cm, is short. Is David really a short man or we have is short. Is David really a short man or we have just drawn an arbitrary line in the sand?just drawn an arbitrary line in the sand? FUZZY LOGIC reflects how people think. It attempts FUZZY LOGIC reflects how people think. It attempts to model our sense of words, our decision making to model our sense of words, our decision making and our common sense.

4 As a result, it is leading to and our common sense. As a result, it is leading to new, more human, intelligent , more human, intelligent In 1965 In 1965 LotfiLotfiZadehZadeh, published his famous paper , published his famous paper FUZZY sets . FUZZY sets . ZadehZadehextended the work on possibility theory into extended the work on possibility theory into a formal system of mathematical LOGIC , and a formal system of mathematical LOGIC , and introduced a new concept for applying natural introduced a new concept for applying natural language terms. language terms. This new LOGIC for representing and manipulating This new LOGIC for representing and manipulating FUZZY terms was called FUZZY terms was called FUZZY logicfuzzy LOGIC , and , and ZadehZadehbecame the became the Master/Father Master/Father of of FUZZY logicfuzzy Why FUZZY ?

5 Why FUZZY ?As As ZadehZadehsaid, the term is concrete, immediate and said, the term is concrete, immediate and descriptive. descriptive. However, many people in the West However, many people in the West were repelled by the word were repelled by the word fuzzyfuzzy, because it is , because it is usually used in a negative used in a negative sense. Why LOGIC ?Why LOGIC ?Fuzziness rests on FUZZY set theory, and FUZZY LOGIC Fuzziness rests on FUZZY set theory, and FUZZY LOGIC is just a small part of that theory. is just a small part of that theory. ZadehZadehused the used the term FUZZY LOGIC in a broader FUZZY LOGIC in a broader LOGIC is a set of mathematical principles FUZZY LOGIC is a set of mathematical principles for knowledge representation based on degrees for knowledge representation based on degrees of membership of membership rather than on crisp membership of rather than on crisp membership of classical binary logicclassical binary twoUnlike two--valued Boolean LOGIC , FUZZY LOGIC is valued Boolean LOGIC , FUZZY LOGIC is multimulti--valuedvalued.

6 It deals with . It deals with degrees of degrees of membershipmembershipand and degrees of truthdegrees of truth.. What is FUZZY LOGIC ?What is FUZZY LOGIC ?8 Range of logical values in Range of logical values in Boolean and FUZZY logicBoolean and FUZZY LOGIC (a) Boolean LOGIC .(b) Multi-valued LOGIC uses the continuum of logical values between 0 FUZZY LOGIC uses the continuum of logical values between 0 (completely false) and 1 (completely true). (completely false) and 1 (completely true). Instead of just black and white, it employs the spectrum of Instead of just black and white, it employs the spectrum of colours, accepting that things can be partly true and partly colours, accepting that things can be partly true and partly false at the same at the same setsFuzzy sets The concept of a The concept of a setsetis fundamental to mathematics.

7 Is fundamental to mathematics. Crisp set theory is governed by a LOGIC that uses one of Crisp set theory is governed by a LOGIC that uses one of only two values: true or false. only two values: true or false. This LOGIC cannot represent vague concepts, and This LOGIC cannot represent vague concepts, and therefore fails to give the answers on the fails to give the answers on the paradoxes. In FUZZY set theory: an element is with a certain degree In FUZZY set theory: an element is with a certain degree of membership. of membership. Thus, a proposition is not either true or false, but Thus, a proposition is not either true or false, but may be partly true (or partly false) to any degree.

8 May be partly true (or partly false) to any degree. This degree is usually taken as a real number in the This degree is usually taken as a real number in the interval [0,1].interval [0,1].10 The classical example in FUZZY sets is The classical example in FUZZY sets is tall mentall men. The . The elements of the FUZZY set tall men are all men, elements of the FUZZY set tall men are all men, but their degrees of membership depend on their but their degrees of membership depend on their height. height. D e g r e e o f M e m b e r s h ipF u z z yM a r kJ o h nT o mB o bB ill111001 . 0 01 . 0 00 . 9 80 . 8 20 . 7 8P e te rS te v e nM ik eD a v idC h r isC r is p100000.

9 2 40 . 1 50 . 0 60 . 0 10 . 0 0N a m eH e ig h t, c m2 0 51 9 81 8 11 6 71 5 51 5 21 5 81 7 21 7 92 0 811D e g r e e o f M e m b e r s h ipF u z z yM a r kJ o h nT o mB o bB ill111001 . 0 01 . 0 00 . 9 80 . 8 20 . 7 8P e te rS te v e nM ik eD a v idC h r isC r is p100000 . 2 40 . 1 50 . 0 60 . 0 10 . 0 0N a m eH e ig h t, c m2 0 51 9 81 8 11 6 71 5 51 5 21 5 81 7 21 7 92 0 8 Crisp set asks the question: Is the man tall?Crisp set asks the question: Is the man tall? Tall men are above 180, and not tall men are below men are above 180, and not tall men are below 180. FUZZY set asks the question: How tall is the man? FUZZY set asks the question: How tall is the man?

10 The tall is partial membership in the FUZZY set, Tom is The tall is partial membership in the FUZZY set, Tom is , cmDegree ofMembershipTa ll Me , cmFuzzy Se tsCrisp SetsCrisp and FUZZY sets of Crisp and FUZZY sets of tall mentall men 13 The The xx--axis represents the axis represents the universe of discourseuniverse of discourse the range of the range of all possible values applicable to a chosen possible values applicable to a chosen variable. The variable is the man s height. According to this The variable is the man s height. According to this representation, the universe of men s heights consists of all tall representation, the universe of men s heights consists of all tall The The yy--axis represents the axis represents the membership value of the FUZZY setmembership value of the FUZZY The FUZZY set of The FUZZY set of tall mentall men maps height values into maps height values into corresponding membership membership sets of FUZZY sets of tall mentall men 14 Let Let XXbe the be the universe of discourseuniverse of discourseand its elements be denoted and its elements be denoted as as xx.