1. Fuzzy sets, fuzzy relational calculus, linguistic approximation
|
|
- Audra Robbins
- 6 years ago
- Views:
Transcription
1 1. Fuzzy sets, fuzzy relational calculus, linguistic approximation 1.1. Fuzzy sets Let us consider a classical set U (Universum) and a real function : U --- L. As a fuzzy set A we understand a set of pairs (x, A (x)) : A = (x, A (x)) A (x) L, x U. (1.4) Function A is called a membership function and is L is usually L = 0,1. Note: If L s degenerated to its border points 0,1 then A is a characteristic function and fuzzy set A is an ordinal set of ordered pairs. Then we speak about a "crisp" set and about a "crisp" value of a membership function. Example 1.4: The expression of fuzzy set by enumeration of its elements (extensional), i.e., by ordered pairs (x, A (x)), x U: A = 0.5/x 1, 0.8/x 2, 0.0/x 3,, 1.0/x k, 0.7/x n, The introduced form of expression for A was often used in early works of prof. Lofti Zadeh. However it is only expressioning convention (x, A (x)) = ( A (x)/x). The advantage of extensional expression is this: it is explicitly given which members belongs to fuzzy set and with which quantity of membership function (e.g., x 2 belongs to U with 80 % certainty; on the contrary x 3 does not belong to U). Height of fuzzy set A - hgt(a): hgt(a) = sup A (x 1 ),, A (x n ), (2.4) x 1,..., x n U Normal fuzzy set: U is ordered set of real numbers, hgt(a) = 1.
2 Normal and a convex fuzzy number: x 1,x 2 U, 0,1, A ( x 1 + (1- ) x 2 ) min A (x 1 ), A (x 2 ), (3.4) Ex: In Fig is a normal convex fuzzy number approximately 9. A as an - cut of fuzzy set A: For 0,1 and ordered subset of U: A = (x, A (x)) A (x), x U, (4.4) E.g. A 0.85 = (8.0, 0.85), (8.5, 0.9), (9.0, 1.0), (9.5, 0.9),... - Fig.1.4. Fig.1.4. Note: Graphs of membership functions may have very strange shapes (bell-functions, sigmoid functions, ). However in case when they represent some language terms they are not absolutely optional. E.g., in case when they represent intensities of some properties that are measured on universum U of real numbers the most of such functions has non monotonic form. (E.g., when we say that the consumption of gasoline of our car is standard then we mean that for interval (0, 3 ) [l/100 km] the value of membership function is zero, then is gradually growing till 5 [l/100 km], where it has value 1, it continues constant till value 6.5 [l/100 km] and then is decreasing till zero in 10 [l/100 km]. Second point of view is the simplicity of the membership function shape. The shapes could be acceptable for manipulations and they would be derived from some not too large library. 91
3 In addition of known bell and sigmoid functions is often used quadruplet notation (parameterization) a,b,, - see. Fig , for x a - -1 (x - a + ), for x a-,a A (x) = 1, for x a,b (5.4) -1 (b + - x), for x b,b+ 0, for x b+. A (x) 1 a- a b b + x U Fig Basic operation with fuzzy sets Let us consider fuzzy sets A, B, C defined gradually on universes U 1, U 2, U 1 U 2 and let us assume L = 0,1. Binary operations union ( ), intersection ( ), complement (Compl) and set subtraction (-) are described by the following relations: C = A B = (x, A B (x)) x (U 1 U 2 ), A B (x) = A (x) (x, A B (x)) x (U 2 U 1 ), A B (x) = B (x) (6.4) (x, A B (x)) x (U 1 U 2 ), A B (x) = max A (x), B (x). C = A B = (x, A B (x)) x (U 1 U 2 ), A B (x) = min A (x), B (x). (7.4) C = Compl A = (x, Compl A (x)) Compl A (x) = 1 - A (x), x U 1. (8.4) 92
4 C = A B = (x, A-B (x)) x U 1, A-B (x) = min A (x),1- B (x). (9.4) Example 2.4: Let us consider the following fuzzy sets A, B and let us construct complement (Compl A), union C = A B and the difference C = A B. A = 0.5/x 1, 0.8/x 2, 0.0/x 3, 0.5/x 4, 0.6/x 5, 1.0/x 6, 0.7/x 7, B = 0.6/y 1, 0.7/y 2, 0.3/x 3, 0.8/x 4, 0.7/x 5, 0.5/y 6, 0.0/y 7, C = A B = 0.5/x 1, 0.8/x 2, 0.2/x 4, 0.3/x 5, 1.0/x 6, 0.7/x 7. x 1 o x 2 o x 6 o x 7 o A x 5 o x 4 o o x 3 o y 6 B o y 2 o y 1 y 7 o Fig. 3.4 Theory of fuzzy sets allows introducing completely new algebraic operations by means so called induction principle: Let us consider fuzzy sets A 1,..., A n defined on universes U 1,...,U n by means of membership functions A1,..., An and n-ary function f, f : U 1 x... x U n Fuzzy set B on V (resp. Its membership function B ) is then induced following way: --- V. by the x 1,...,x n U 1 x... x U n, y V, f(x 1,...,x n ) = y Sup min A1 (x 1 ),..., An (x n ), pro f -1 (V) x 1,..., x n U 1 x... x U n B (y) = (10.4) 0, pro f -1 (V) =. Example 3.4: Let us consider fuzzy numbers A1 = 2, 2, 1, 1 a A2 = 4, 4, 1.5, 1 defined on universum R 1 and function f: R 1 R 1 R 1 defined as an ordinary function 93
5 of addition, i.e., f(x 1, x 2 ) = x 1 + x 2. The application of expression (10.4) is for the main quantities introduced in the following table: Row X 1 x 2 y A1 (x 1 ) A2 (x 2 ) B (y) The expressions of application of (10.4) introduced in the table is possible to compare with results of arithmetic operation + (from the section 1.2.1). By the Induction principle is very easy to define the set of algebraic operations f F on the set of fuzzy sets or for the definition of so called approximation of fuzzy sets Let us consider fuzzy sets A, A 1,..., A n A, the operation f F appropriate metric function d: 0,1 x 0, ,1. As an approximation of the set B, that is formed as a result of function f, B = f(a 1,..., A n ), is understood a fuzzy set A, for which holds: and some d( B (x), A (x)) = min d( B (x), C (x)). (11.4) C A By application of induction principle on concept of multiplication of fuzzy relations is possible to construct new fuzzy relations: Let us consider classical Relations R, S (for the simplicity let us consider Binary relations) defined on Cartesian products of universes U 1, U 2, U 3, i.e., R U 1 x U 2, S U 2 x U 3. Fuzzy relations A R, A S are defined as fuzzy sets on sets R, S. The product of fuzzy relations A R, A S is defined by means of induction principle: RoS (x 1, x 3 ) = Sup min AR (x 1, x 2 ), AS (x 2, x 3 ), (12.4) (x 1,x 2,x 3 ) U 1 x U 2 x U 3 94
6 where RoS is the product of relations R, S. By means of (12.4) it is very simply to define some other useful properties, as it is, e.g., transitivity of fuzzy ordering (fuzzy transitivity): R S, U x U, R o R = R, A R = A S, (x 1,x 2,x 3 ) U x U x U o (x 1, x 3 ) = max min (x 1, x 2 ), (x 2, x 3 ), (13.4) (x 1, x 2, x 3 ) U x U x U Example 4.4.: Let us consider universum U = a, b, c, d on which is defined fuzzy relation ordering by the following way: x, y U, (x,y) = 0.1/(a b), 0.5/(a d), 0.4/(b c), 0.9/(c d). (14.4) Membership function (x,y) is expressed by individual pairs where the number above the symbol "/" gives the certainty of the statement that is given below "/". The computation of membership function value for the remaining two pairs on U - ((a,c), (b,d)) uses the transitivity of relation and its description by (14.4). Applying (8.4) x,y U, (y,x) = 1 - (x,y), (15.4) we obtain values (a,c) a (b,d). (a,c) = max min (a,b), (b,c), min (a,d), (d,c) = = max min 0.1, 0.4, min 0.5, 0.1 = max 0.1, 0.1 = 0.1. (b,d) = max min (b,c), (c,d), min (b,a), (a,d = = max min 0.4, 0.9, min 0.5, 0.5 = max 0.4, 0.5 =
7 1.2.1 A basic arithmetics of parametrically defined fuzzy numbers Let us consider two fuzzy numbers Q(x) = a,b,, = m, Q(y) = c,d,, = n. The constructions of arithmetic operations f are given in Table 1.4. The symbol 0 is the cut by a partial ordering for = 0. Tab.1.4. Operation f Computation formula Conditions n ( d, c,, ) n,, ( d ), ( c ) c n 0 0, n 0 d c d 0 m n ( a c, b d,, ) m n ( a d, b c,, ) ( ac, bd, a c, b d ) m, n 0 m n , n 0 0 0, n 0 0 0, n 0 ( ad, bc, d a, b c ) m 0 ( bc, ad, b c, d a ) m 0 ( bd, ac, b d, a c ) m Linguistic variable and the space of fuzzy values Though we have studied the above concepts (and operations with them) in general levels (similarly as in mathematics, where we are not interested in units of contents and variables) in the development of the space of fuzzy values in real cases is always considered a certain system or real environment. In other words spaces of fuzzy values are semantically bonded. This fact is included in the concept of linguistic variable that contains with fuzzy space also the name of the variable and the structuralisation of participating universes. As a linguistic variable LV is understood a triplet LV = J, A, U, (16.4) where J is the name of the variable (e.g., the temperature in Prague in June"), A is the set fuzzy linguistic values of the variable J (e.g., under long time standard (PDN), low (Nízká), middle (Střední),"above long time standard (NDN)",...) and U is the structure of universes, on which are the linguistic variable and its fuzzy 96
8 values defined. (In our case - "the temperature in Prague in June is a universum of temperature scale (however it could be in general Cartesian product of some sub universes U = U 1 x... x U n.)) In Fig is illustrated linguistic variable "T 6 temperature in Prague in June". Fig In the context of application of fuzzy sets for modeling of systems is very natural to assume that exists a certain intuitive image about a term system variable (temperature, density, aggresivity, ). By means of these variables is possible to describe the system and its behavior qualitatively, e.g., on the base of observation (without an exact numerical model) in terms of qualitative values and their qualitative derivations (temperature is high, the density is low, the temperature is increasing, the aggresivity is decreasing, ). Fuzzy qualitative value may be then understood as a fuzzy number with certain semantic content. The set of all qualitative values related to all considered variables of the modeled system is called as a space of fuzzy values - Q F. Mainly for technical purposes was done a certain unification of fuzzy spaces. One of examples of fuzzy spaces is in Fig (for case U = -1,1 ). Q F = - Velká (Large), - Střední (Middle), - Malá (Small), Nula (Zero), + Malá (Small), + Střední (Middle), + Velká (Large) Fig
9 1.4. Linguistic approximation With help of operations from Table 1.4. we may compute values of fuzzy variables in space Q F. Results are new fuzzy values B (fuzzy numbers), that might not be none value from Q F. However no inter-values in Q F we have. For the linguistic determination of computed number we need special operation, so called linguistic approximation. This operation select from the values in Q F the nearest (A) value to B. (Speaking about distance we need some metric function d(b, A).) The method of linguistic approximation will be now explained by example for computed fuzzy number Q(z) = c,d,, = B, its approximation A = a,b,, and metric function (17.4). d(b, A) = 1/2 2(a-b-c+d) /2 c + d - a - b 2 1/2, (17.4) Example 5.4.: Let us consider the space Q F given in Fig.5.4 with the following fuzzy numbers: Nula (Zero) = 0, 0, 0, 0, Malá (Small) = 0, 0.2, 0, 0.2, Střední (Middle) = 0.4, 0.7, 0.1, 0.2, Velká (Large) = 0.9, 1, 0.1, 0, Determine semantic content of fuzzy number B = Small + Large. a) Arithmetic operation + from Table 1.4: B = p, q, r, s + j, l, m, n = p + j, q + l, r + m, s + n B = 0, 0.2, 0, , 1, 0.1, 0 = 0.9, 1.2, 0.1, 0.2 = B. b) Linguistic approximation using (17.4): We have to compute: d(b, Zero) =, d(b, Small) =, d(b, Middle) = 0.5, d(b, Large) =
10 Result: Small + Large Large. Check examples: 1) For fuzzy value "NDN" of varibale Temperature in Prague in June construct the cut for = 0.75 by means of enumeration of some quantities. 2) Find at least two cuts (in Fig. 4.4) for which holds value (A1) = value (A2). 3) Let us consider fuzzy sets A, B on universum U = x 1,...,x 4 defined by enumeration of some elements: A = 0.32/x 1, 0.45/x 2, 0.63/x 3, 0.11/x 4, B = 0.75/x 1, 0.12/x 2, 0.27/x 3, 0.81/x 4. Construct fuzzy set C = Compl (A B)! 4) Let us consider universum U = a,b,c,d, on which is defined fuzzy relation equivalence " " by the following way: x,y U, (x,y) = 0.1/(a b), 0.5/(a d), 0.4/(b c), 0.9/(c d). (Relation equivalence has the property «transitivity» as relation ordering and it is possible to use expressions (12.4), (13.4) and method for computation of relation of order for pairs (a,c), (b,d). Equivalence is symmetric and instead of (15.4) is necessary to use x,y U, (x, y) = (y, x). 5) Let us consider the space Q F from Figure obr Compute linguistic approximation B = Small Middle. 99
11 The list of recommended literature: [1] Zadeh, L., A: Fuzzy Sets. Information and Control. No.8., 1965, str [2] Leitch, R. and Shen, Q.: Fuzzy Qualitative Simulation. IEEE Trans. on Syst., Man and Cybernet., No.4., str [3] Sugeno, M. and Yasukawa, T.: A Fuzzy-Logic-Based Approach to Qualitative Modelling. Trans. on Fuzzy Systems, Vol. 1., No. 1., str [4] Bíla, J., Šmíd, J., Král, F. a Hlaváč, V.: Informační technologie. Databázové a znalostní systémy. ČVUT v Praze,
CHAPTER 5 FUZZY LOGIC CONTROL
64 CHAPTER 5 FUZZY LOGIC CONTROL 5.1 Introduction Fuzzy logic is a soft computing tool for embedding structured human knowledge into workable algorithms. The idea of fuzzy logic was introduced by Dr. Lofti
More informationApproximation of Multiplication of Trapezoidal Epsilon-delta Fuzzy Numbers
Advances in Fuzzy Mathematics ISSN 973-533X Volume, Number 3 (7, pp 75-735 Research India Publications http://wwwripublicationcom Approximation of Multiplication of Trapezoidal Epsilon-delta Fuzzy Numbers
More informationFuzzy Sets and Systems. Lecture 1 (Introduction) Bu- Ali Sina University Computer Engineering Dep. Spring 2010
Fuzzy Sets and Systems Lecture 1 (Introduction) Bu- Ali Sina University Computer Engineering Dep. Spring 2010 Fuzzy sets and system Introduction and syllabus References Grading Fuzzy sets and system Syllabus
More informationApplication of fuzzy set theory in image analysis. Nataša Sladoje Centre for Image Analysis
Application of fuzzy set theory in image analysis Nataša Sladoje Centre for Image Analysis Our topics for today Crisp vs fuzzy Fuzzy sets and fuzzy membership functions Fuzzy set operators Approximate
More informationFUZZY SETS. Precision vs. Relevancy LOOK OUT! A 1500 Kg mass is approaching your head OUT!!
FUZZY SETS Precision vs. Relevancy A 5 Kg mass is approaching your head at at 45.3 45.3 m/sec. m/s. OUT!! LOOK OUT! 4 Introduction How to simplify very complex systems? Allow some degree of uncertainty
More informationIntroduction to Fuzzy Logic. IJCAI2018 Tutorial
Introduction to Fuzzy Logic IJCAI2018 Tutorial 1 Crisp set vs. Fuzzy set A traditional crisp set A fuzzy set 2 Crisp set vs. Fuzzy set 3 Crisp Logic Example I Crisp logic is concerned with absolutes-true
More informationApproximate Reasoning with Fuzzy Booleans
Approximate Reasoning with Fuzzy Booleans P.M. van den Broek Department of Computer Science, University of Twente,P.O.Box 217, 7500 AE Enschede, the Netherlands pimvdb@cs.utwente.nl J.A.R. Noppen Department
More informationNotes on Fuzzy Set Ordination
Notes on Fuzzy Set Ordination Umer Zeeshan Ijaz School of Engineering, University of Glasgow, UK Umer.Ijaz@glasgow.ac.uk http://userweb.eng.gla.ac.uk/umer.ijaz May 3, 014 1 Introduction The membership
More informationARTIFICIAL INTELLIGENCE. Uncertainty: fuzzy systems
INFOB2KI 2017-2018 Utrecht University The Netherlands ARTIFICIAL INTELLIGENCE Uncertainty: fuzzy systems Lecturer: Silja Renooij These slides are part of the INFOB2KI Course Notes available from www.cs.uu.nl/docs/vakken/b2ki/schema.html
More informationFigure-12 Membership Grades of x o in the Sets A and B: μ A (x o ) =0.75 and μb(xo) =0.25
Membership Functions The membership function μ A (x) describes the membership of the elements x of the base set X in the fuzzy set A, whereby for μ A (x) a large class of functions can be taken. Reasonable
More informationCHAPTER 3 FUZZY RELATION and COMPOSITION
CHAPTER 3 FUZZY RELATION and COMPOSITION Crisp relation! Definition (Product set) Let A and B be two non-empty sets, the prod uct set or Cartesian product A B is defined as follows, A B = {(a, b) a A,
More informationReview of Fuzzy Logical Database Models
IOSR Journal of Computer Engineering (IOSRJCE) ISSN: 2278-0661, ISBN: 2278-8727Volume 8, Issue 4 (Jan. - Feb. 2013), PP 24-30 Review of Fuzzy Logical Database Models Anupriya 1, Prof. Rahul Rishi 2 1 (Department
More informationCHAPTER 4 FREQUENCY STABILIZATION USING FUZZY LOGIC CONTROLLER
60 CHAPTER 4 FREQUENCY STABILIZATION USING FUZZY LOGIC CONTROLLER 4.1 INTRODUCTION Problems in the real world quite often turn out to be complex owing to an element of uncertainty either in the parameters
More informationCHAPTER 3 FUZZY RELATION and COMPOSITION
CHAPTER 3 FUZZY RELATION and COMPOSITION The concept of fuzzy set as a generalization of crisp set has been introduced in the previous chapter. Relations between elements of crisp sets can be extended
More informationChapter 2: FUZZY SETS
Ch.2: Fuzzy sets 1 Chapter 2: FUZZY SETS Introduction (2.1) Basic Definitions &Terminology (2.2) Set-theoretic Operations (2.3) Membership Function (MF) Formulation & Parameterization (2.4) Complement
More informationFuzzy Set, Fuzzy Logic, and its Applications
Sistem Cerdas (TE 4485) Fuzzy Set, Fuzzy Logic, and its pplications Instructor: Thiang Room: I.201 Phone: 031-2983115 Email: thiang@petra.ac.id Sistem Cerdas: Fuzzy Set and Fuzzy Logic - 1 Introduction
More informationIntroduction to Fuzzy Logic and Fuzzy Systems Adel Nadjaran Toosi
Introduction to Fuzzy Logic and Fuzzy Systems Adel Nadjaran Toosi Fuzzy Slide 1 Objectives What Is Fuzzy Logic? Fuzzy sets Membership function Differences between Fuzzy and Probability? Fuzzy Inference.
More informationUnit V. Neural Fuzzy System
Unit V Neural Fuzzy System 1 Fuzzy Set In the classical set, its characteristic function assigns a value of either 1 or 0 to each individual in the universal set, There by discriminating between members
More informationFUNDAMENTALS OF FUZZY SETS
FUNDAMENTALS OF FUZZY SETS edited by Didier Dubois and Henri Prade IRIT, CNRS & University of Toulouse III Foreword by LotfiA. Zadeh 14 Kluwer Academic Publishers Boston//London/Dordrecht Contents Foreword
More informationWhat is all the Fuzz about?
What is all the Fuzz about? Fuzzy Systems CPSC 433 Christian Jacob Dept. of Computer Science Dept. of Biochemistry & Molecular Biology University of Calgary Fuzzy Systems in Knowledge Engineering Fuzzy
More informationMODELING FOR RESIDUAL STRESS, SURFACE ROUGHNESS AND TOOL WEAR USING AN ADAPTIVE NEURO FUZZY INFERENCE SYSTEM
CHAPTER-7 MODELING FOR RESIDUAL STRESS, SURFACE ROUGHNESS AND TOOL WEAR USING AN ADAPTIVE NEURO FUZZY INFERENCE SYSTEM 7.1 Introduction To improve the overall efficiency of turning, it is necessary to
More informationIntroduction 3 Fuzzy Inference. Aleksandar Rakić Contents
Beograd ETF Fuzzy logic Introduction 3 Fuzzy Inference Aleksandar Rakić rakic@etf.rs Contents Mamdani Fuzzy Inference Fuzzification of the input variables Rule evaluation Aggregation of rules output Defuzzification
More informationFuzzy Sets and Systems. Lecture 2 (Fuzzy Sets) Bu- Ali Sina University Computer Engineering Dep. Spring 2010
Fuzzy Sets and Systems Lecture 2 (Fuzzy Sets) Bu- Ali Sina University Computer Engineering Dep. Spring 2010 Fuzzy Sets Formal definition: A fuzzy set A in X (universal set) is expressed as a set of ordered
More informationFuzzy Logic. This amounts to the use of a characteristic function f for a set A, where f(a)=1 if the element belongs to A, otherwise it is 0;
Fuzzy Logic Introduction: In Artificial Intelligence (AI) the ultimate goal is to create machines that think like humans. Human beings make decisions based on rules. Although, we may not be aware of it,
More informationROUGH MEMBERSHIP FUNCTIONS: A TOOL FOR REASONING WITH UNCERTAINTY
ALGEBRAIC METHODS IN LOGIC AND IN COMPUTER SCIENCE BANACH CENTER PUBLICATIONS, VOLUME 28 INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WARSZAWA 1993 ROUGH MEMBERSHIP FUNCTIONS: A TOOL FOR REASONING
More informationGEOG 5113 Special Topics in GIScience. Why is Classical set theory restricted? Contradiction & Excluded Middle. Fuzzy Set Theory in GIScience
GEOG 5113 Special Topics in GIScience Fuzzy Set Theory in GIScience -Basic Properties and Concepts of Fuzzy Sets- Why is Classical set theory restricted? Boundaries of classical sets are required to be
More informationFuzzy Rules & Fuzzy Reasoning
Sistem Cerdas : PTK Pasca Sarjana - UNY Fuzzy Rules & Fuzzy Reasoning Pengampu: Fatchul rifin Referensi: Jyh-Shing Roger Jang et al., Neuro-Fuzzy and Soft Computing: Computational pproach to Learning and
More informationANALYSIS AND REASONING OF DATA IN THE DATABASE USING FUZZY SYSTEM MODELLING
ANALYSIS AND REASONING OF DATA IN THE DATABASE USING FUZZY SYSTEM MODELLING Dr.E.N.Ganesh Dean, School of Engineering, VISTAS Chennai - 600117 Abstract In this paper a new fuzzy system modeling algorithm
More informationX : U -> [0, 1] R : U x V -> [0, 1]
A Fuzzy Logic 2000 educational package for Mathematica Marian S. Stachowicz and Lance Beall Electrical and Computer Engineering University of Minnesota Duluth, Minnesota 55812-2496, USA http://www.d.umn.edu/ece/lis
More informationXI International PhD Workshop OWD 2009, October Fuzzy Sets as Metasets
XI International PhD Workshop OWD 2009, 17 20 October 2009 Fuzzy Sets as Metasets Bartłomiej Starosta, Polsko-Japońska WyŜsza Szkoła Technik Komputerowych (24.01.2008, prof. Witold Kosiński, Polsko-Japońska
More informationFuzzy Logic : Introduction
Fuzzy Logic : Introduction Debasis Samanta IIT Kharagpur dsamanta@iitkgp.ac.in 23.01.2018 Debasis Samanta (IIT Kharagpur) Soft Computing Applications 23.01.2018 1 / 69 What is Fuzzy logic? Fuzzy logic
More informationWhat is all the Fuzz about?
What is all the Fuzz about? Fuzzy Systems: Introduction CPSC 533 Christian Jacob Dept. of Computer Science Dept. of Biochemistry & Molecular Biology University of Calgary Fuzzy Systems in Knowledge Engineering
More informationA fuzzy subset of a set A is any mapping f : A [0, 1], where [0, 1] is the real unit closed interval. the degree of membership of x to f
Algebraic Theory of Automata and Logic Workshop Szeged, Hungary October 1, 2006 Fuzzy Sets The original Zadeh s definition of a fuzzy set is: A fuzzy subset of a set A is any mapping f : A [0, 1], where
More informationChapter 4 Fuzzy Logic
4.1 Introduction Chapter 4 Fuzzy Logic The human brain interprets the sensory information provided by organs. Fuzzy set theory focus on processing the information. Numerical computation can be performed
More informationVague Congruence Relation Induced by VLI Ideals of Lattice Implication Algebras
American Journal of Mathematics and Statistics 2016, 6(3): 89-93 DOI: 10.5923/j.ajms.20160603.01 Vague Congruence Relation Induced by VLI Ideals of Lattice Implication Algebras T. Anitha 1,*, V. Amarendra
More informationChapter 7 Fuzzy Logic Controller
Chapter 7 Fuzzy Logic Controller 7.1 Objective The objective of this section is to present the output of the system considered with a fuzzy logic controller to tune the firing angle of the SCRs present
More informationOn JAM of Triangular Fuzzy Number Matrices
117 On JAM of Triangular Fuzzy Number Matrices C.Jaisankar 1 and R.Durgadevi 2 Department of Mathematics, A. V. C. College (Autonomous), Mannampandal 609305, India ABSTRACT The fuzzy set theory has been
More informationA.1 Numbers, Sets and Arithmetic
522 APPENDIX A. MATHEMATICS FOUNDATIONS A.1 Numbers, Sets and Arithmetic Numbers started as a conceptual way to quantify count objects. Later, numbers were used to measure quantities that were extensive,
More informationTHREE LECTURES ON BASIC TOPOLOGY. 1. Basic notions.
THREE LECTURES ON BASIC TOPOLOGY PHILIP FOTH 1. Basic notions. Let X be a set. To make a topological space out of X, one must specify a collection T of subsets of X, which are said to be open subsets of
More informationProperties of a Function s Graph
Section 3.2 Properties of a Function s Graph Objective 1: Determining the Intercepts of a Function An intercept of a function is a point on the graph of a function where the graph either crosses or touches
More informationNeural Networks Lesson 9 - Fuzzy Logic
Neural Networks Lesson 9 - Prof. Michele Scarpiniti INFOCOM Dpt. - Sapienza University of Rome http://ispac.ing.uniroma1.it/scarpiniti/index.htm michele.scarpiniti@uniroma1.it Rome, 26 November 2009 M.
More informationRelational Database: The Relational Data Model; Operations on Database Relations
Relational Database: The Relational Data Model; Operations on Database Relations Greg Plaxton Theory in Programming Practice, Spring 2005 Department of Computer Science University of Texas at Austin Overview
More informationFuzzy If-Then Rules. Fuzzy If-Then Rules. Adnan Yazıcı
Fuzzy If-Then Rules Adnan Yazıcı Dept. of Computer Engineering, Middle East Technical University Ankara/Turkey Fuzzy If-Then Rules There are two different kinds of fuzzy rules: Fuzzy mapping rules and
More informationWalheer Barnabé. Topics in Mathematics Practical Session 2 - Topology & Convex
Topics in Mathematics Practical Session 2 - Topology & Convex Sets Outline (i) Set membership and set operations (ii) Closed and open balls/sets (iii) Points (iv) Sets (v) Convex Sets Set Membership and
More informationIPMU July 2-7, 2006 Paris, France
IPMU July 2-7, 2006 Paris, France Information Processing and Management of Uncertainty in Knowledge-Based Systems Conceptual Design and Implementation of the Salem Chakhar 1 and Abelkader Telmoudi 2 1
More informationThe Travelling Salesman Problem. in Fuzzy Membership Functions 1. Abstract
Chapter 7 The Travelling Salesman Problem in Fuzzy Membership Functions 1 Abstract In this chapter, the fuzzification of travelling salesman problem in the way of trapezoidal fuzzy membership functions
More informationA Brief Idea on Fuzzy and Crisp Sets
International OPEN ACCESS Journal Of Modern Engineering Research (IJMER) A Brief Idea on Fuzzy and Crisp Sets Rednam SS Jyothi 1, Eswar Patnala 2, K.Asish Vardhan 3 (Asst.Prof(c),Information Technology,
More informationLinguistic Values on Attribute Subdomains in Vague Database Querying
Linguistic Values on Attribute Subdomains in Vague Database Querying CORNELIA TUDORIE Department of Computer Science and Engineering University "Dunărea de Jos" Domnească, 82 Galaţi ROMANIA Abstract: -
More informationCHAPTER 4 FUZZY LOGIC, K-MEANS, FUZZY C-MEANS AND BAYESIAN METHODS
CHAPTER 4 FUZZY LOGIC, K-MEANS, FUZZY C-MEANS AND BAYESIAN METHODS 4.1. INTRODUCTION This chapter includes implementation and testing of the student s academic performance evaluation to achieve the objective(s)
More informationIntroduction 2 Fuzzy Sets & Fuzzy Rules. Aleksandar Rakić Contents
Beograd ETF Fuzzy logic Introduction 2 Fuzzy Sets & Fuzzy Rules Aleksandar Rakić rakic@etf.rs Contents Characteristics of Fuzzy Sets Operations Properties Fuzzy Rules Examples 2 1 Characteristics of Fuzzy
More informationFuzzy Set-Theoretical Approach for Comparing Objects with Fuzzy Attributes
Fuzzy Set-Theoretical Approach for Comparing Objects with Fuzzy Attributes Y. Bashon, D. Neagu, M.J. Ridley Department of Computing University of Bradford Bradford, BD7 DP, UK e-mail: {Y.Bashon, D.Neagu,
More informationIT 201 Digital System Design Module II Notes
IT 201 Digital System Design Module II Notes BOOLEAN OPERATIONS AND EXPRESSIONS Variable, complement, and literal are terms used in Boolean algebra. A variable is a symbol used to represent a logical quantity.
More informationSETS. Sets are of two sorts: finite infinite A system of sets is a set, whose elements are again sets.
SETS A set is a file of objects which have at least one property in common. The objects of the set are called elements. Sets are notated with capital letters K, Z, N, etc., the elements are a, b, c, d,
More informationFACILITY LIFE-CYCLE COST ANALYSIS BASED ON FUZZY SETS THEORY Life-cycle cost analysis
FACILITY LIFE-CYCLE COST ANALYSIS BASED ON FUZZY SETS THEORY Life-cycle cost analysis J. O. SOBANJO FAMU-FSU College of Engineering, Tallahassee, Florida Durability of Building Materials and Components
More informationFuzzy logic. 1. Introduction. 2. Fuzzy sets. Radosªaw Warzocha. Wrocªaw, February 4, Denition Set operations
Fuzzy logic Radosªaw Warzocha Wrocªaw, February 4, 2014 1. Introduction A fuzzy concept appearing in works of many philosophers, eg. Hegel, Nietzche, Marx and Engels, is a concept the value of which can
More informationCS 1200 Discrete Math Math Preliminaries. A.R. Hurson 323 CS Building, Missouri S&T
CS 1200 Discrete Math A.R. Hurson 323 CS Building, Missouri S&T hurson@mst.edu 1 Course Objective: Mathematical way of thinking in order to solve problems 2 Variable: holder. A variable is simply a place
More informationComputational Intelligence Lecture 12:Linguistic Variables and Fuzzy Rules
Computational Intelligence Lecture 12:Linguistic Variables and Fuzzy Rules Farzaneh Abdollahi Department of Electrical Engineering Amirkabir University of Technology Fall 2011 Farzaneh Abdollahi Computational
More informationSets 1. The things in a set are called the elements of it. If x is an element of the set S, we say
Sets 1 Where does mathematics start? What are the ideas which come first, in a logical sense, and form the foundation for everything else? Can we get a very small number of basic ideas? Can we reduce it
More informationL. A. Zadeh: Fuzzy Sets. (1965) A review
POSSIBILISTIC INFORMATION: A Tutorial L. A. Zadeh: Fuzzy Sets. (1965) A review George J. Klir Petr Osička State University of New York (SUNY) Binghamton, New York 13902, USA gklir@binghamton.edu Palacky
More informationAssessment of Human Skills Using Trapezoidal Fuzzy Numbers (Part II)
American Journal of Computational and Applied Mathematics 05, 5(5): 54-58 DOI: 0.593/j.ajcam.050505.04 Assessment of Human Skills Using Trapezoidal Fuzzy Numbers (Part II) Michael Gr. Voskoglou Department
More informationIntroduction. The Quine-McCluskey Method Handout 5 January 24, CSEE E6861y Prof. Steven Nowick
CSEE E6861y Prof. Steven Nowick The Quine-McCluskey Method Handout 5 January 24, 2013 Introduction The Quine-McCluskey method is an exact algorithm which finds a minimum-cost sum-of-products implementation
More informationMatrix Inference in Fuzzy Decision Trees
Matrix Inference in Fuzzy Decision Trees Santiago Aja-Fernández LPI, ETSIT Telecomunicación University of Valladolid, Spain sanaja@tel.uva.es Carlos Alberola-López LPI, ETSIT Telecomunicación University
More informationSpeed regulation in fan rotation using fuzzy inference system
58 Scientific Journal of Maritime Research 29 (2015) 58-63 Faculty of Maritime Studies Rijeka, 2015 Multidisciplinary SCIENTIFIC JOURNAL OF MARITIME RESEARCH Multidisciplinarni znanstveni časopis POMORSTVO
More informationApplication of Shortest Path Algorithm to GIS using Fuzzy Logic
Application of Shortest Path Algorithm to GIS using Fuzzy Logic Petrík, S. * - Madarász, L. ** - Ádám, N. * - Vokorokos, L. * * Department of Computers and Informatics, Faculty of Electrical Engineering
More informationThis Lecture. We will first introduce some basic set theory before we do counting. Basic Definitions. Operations on Sets.
Sets A B C This Lecture We will first introduce some basic set theory before we do counting. Basic Definitions Operations on Sets Set Identities Defining Sets Definition: A set is an unordered collection
More informationFuzzy Logic Approach towards Complex Solutions: A Review
Fuzzy Logic Approach towards Complex Solutions: A Review 1 Arnab Acharyya, 2 Dipra Mitra 1 Technique Polytechnic Institute, 2 Technique Polytechnic Institute Email: 1 cst.arnab@gmail.com, 2 mitra.dipra@gmail.com
More informationAbout the Tutorial. Audience. Prerequisites. Copyright & Disclaimer. Discrete Mathematics
About the Tutorial Discrete Mathematics is a branch of mathematics involving discrete elements that uses algebra and arithmetic. It is increasingly being applied in the practical fields of mathematics
More informationFUZZY BOOLEAN ALGEBRAS AND LUKASIEWICZ LOGIC. Angel Garrido
Acta Universitatis Apulensis ISSN: 1582-5329 No. 22/2010 pp. 101-111 FUZZY BOOLEAN ALGEBRAS AND LUKASIEWICZ LOGIC Angel Garrido Abstract. In this paper, we analyze the more adequate tools to solve many
More informationFuzzy Reasoning. Outline
Fuzzy Reasoning Outline Introduction Bivalent & Multivalent Logics Fundamental fuzzy concepts Fuzzification Defuzzification Fuzzy Expert System Neuro-fuzzy System Introduction Fuzzy concept first introduced
More informationCT79 SOFT COMPUTING ALCCS-FEB 2014
Q.1 a. Define Union, Intersection and complement operations of Fuzzy sets. For fuzzy sets A and B Figure Fuzzy sets A & B The union of two fuzzy sets A and B is a fuzzy set C, written as C=AUB or C=A OR
More informationDisjunctive and Conjunctive Normal Forms in Fuzzy Logic
Disjunctive and Conjunctive Normal Forms in Fuzzy Logic K. Maes, B. De Baets and J. Fodor 2 Department of Applied Mathematics, Biometrics and Process Control Ghent University, Coupure links 653, B-9 Gent,
More informationGet Free notes at Module-I One s Complement: Complement all the bits.i.e. makes all 1s as 0s and all 0s as 1s Two s Complement: One s complement+1 SIGNED BINARY NUMBERS Positive integers (including zero)
More informationUniversity of Groningen. Morphological design of Discrete-Time Cellular Neural Networks Brugge, Mark Harm ter
University of Groningen Morphological design of Discrete-Time Cellular Neural Networks Brugge, Mark Harm ter IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you
More informationAssignment (3-6) Boolean Algebra and Logic Simplification - General Questions
Assignment (3-6) Boolean Algebra and Logic Simplification - General Questions 1. Convert the following SOP expression to an equivalent POS expression. 2. Determine the values of A, B, C, and D that make
More informationFuzzy if-then rules fuzzy database modeling
Fuzzy if-then rules Associates a condition described using linguistic variables and fuzzy sets to a conclusion A scheme for capturing knowledge that involves imprecision 23.11.2010 1 fuzzy database modeling
More informationEquations and Functions, Variables and Expressions
Equations and Functions, Variables and Expressions Equations and functions are ubiquitous components of mathematical language. Success in mathematics beyond basic arithmetic depends on having a solid working
More informationFuzzy Mathematics. Fuzzy -Sets, -Relations, -Logic, -Graphs, -Mappings and The Extension Principle. Olaf Wolkenhauer. Control Systems Centre UMIST
Fuzzy Mathematics Fuzzy -Sets, -Relations, -Logic, -Graphs, -Mappings and The Extension Principle Olaf Wolkenhauer Control Systems Centre UMIST o.wolkenhauer@umist.ac.uk www.csc.umist.ac.uk/people/wolkenhauer.htm
More information[Y2] Counting and understanding number. [Y2] Counting and understanding number. [Y2] Counting and understanding number
Medium Term Plan : Year 2 Autumn Term Block A1.a: Count on and back in 1s or 10s from a 2- digit number; write figures up to 100 Block A1.b: Begin to count up to 100 objects by grouping in 5s or 10s; estimate
More informationIntroduction. Aleksandar Rakić Contents
Beograd ETF Fuzzy logic Introduction Aleksandar Rakić rakic@etf.rs Contents Definitions Bit of History Fuzzy Applications Fuzzy Sets Fuzzy Boundaries Fuzzy Representation Linguistic Variables and Hedges
More informationFUZZY INFERENCE. Siti Zaiton Mohd Hashim, PhD
FUZZY INFERENCE Siti Zaiton Mohd Hashim, PhD Fuzzy Inference Introduction Mamdani-style inference Sugeno-style inference Building a fuzzy expert system 9/29/20 2 Introduction Fuzzy inference is the process
More information2 Dept. of Computer Applications 3 Associate Professor Dept. of Computer Applications
International Journal of Computing Science and Information Technology, 2014, Vol.2(2), 15-19 ISSN: 2278-9669, April 2014 (http://ijcsit.org) Optimization of trapezoidal balanced Transportation problem
More informationOptimization with linguistic variables
Optimization with linguistic variables Christer Carlsson christer.carlsson@abo.fi Robert Fullér rfuller@abo.fi Abstract We consider fuzzy mathematical programming problems (FMP) in which the functional
More informationFuzzy Transform for Low-contrast Image Enhancement
Fuzzy Transform for Low-contrast Image Enhancement Reshmalakshmi C. Dept. of Electronics, College of Engineering, Trivandrum, India. Sasikumar M. Dept. of Electronics, Marian Engineering College, Trivandrum,
More informationOn Fuzzy Topological Spaces Involving Boolean Algebraic Structures
Journal of mathematics and computer Science 15 (2015) 252-260 On Fuzzy Topological Spaces Involving Boolean Algebraic Structures P.K. Sharma Post Graduate Department of Mathematics, D.A.V. College, Jalandhar
More information6.1 Combinational Circuits. George Boole ( ) Claude Shannon ( )
6. Combinational Circuits George Boole (85 864) Claude Shannon (96 2) Signals and Wires Digital signals Binary (or logical ) values: or, on or off, high or low voltage Wires. Propagate digital signals
More informationWhy Fuzzy Fuzzy Logic and Sets Fuzzy Reasoning. DKS - Module 7. Why fuzzy thinking?
Fuzzy Systems Overview: Literature: Why Fuzzy Fuzzy Logic and Sets Fuzzy Reasoning chapter 4 DKS - Module 7 1 Why fuzzy thinking? Experts rely on common sense to solve problems Representation of vague,
More informationDiscrete Mathematics. Chapter 7. trees Sanguk Noh
Discrete Mathematics Chapter 7. trees Sanguk Noh Table Trees Labeled Trees Tree searching Undirected trees Minimal Spanning Trees Trees Theorem : Let (T, v ) be a rooted tree. Then, There are no cycles
More informationFuzzy type-2 in Shortest Path and Maximal Flow Problems
Global Journal of Pure and Applied Mathematics. ISSN 0973-1768 Volume 13, Number 9 (2017), pp. 6595-6607 Research India Publications http://www.ripublication.com Fuzzy type-2 in Shortest Path and Maximal
More informationA set with only one member is called a SINGLETON. A set with no members is called the EMPTY SET or 2 N
Mathematical Preliminaries Read pages 529-540 1. Set Theory 1.1 What is a set? A set is a collection of entities of any kind. It can be finite or infinite. A = {a, b, c} N = {1, 2, 3, } An entity is an
More informationCOMBINATION OF ROUGH AND FUZZY SETS
1 COMBINATION OF ROUGH AND FUZZY SETS BASED ON α-level SETS Y.Y. Yao Department of Computer Science, Lakehead University Thunder Bay, Ontario, Canada P7B 5E1 E-mail: yyao@flash.lakeheadu.ca 1 ABSTRACT
More informationDifferent strategies to solve fuzzy linear programming problems
ecent esearch in Science and Technology 2012, 4(5): 10-14 ISSN: 2076-5061 Available Online: http://recent-science.com/ Different strategies to solve fuzzy linear programming problems S. Sagaya oseline
More informationMembership Value Assignment
FUZZY SETS Membership Value Assignment There are possible more ways to assign membership values or function to fuzzy variables than there are to assign probability density functions to random variables
More informationUNC Charlotte 2010 Comprehensive
00 Comprehensive March 8, 00. A cubic equation x 4x x + a = 0 has three roots, x, x, x. If x = x + x, what is a? (A) 4 (B) 8 (C) 0 (D) (E) 6. For which value of a is the polynomial P (x) = x 000 +ax+9
More informationFuzzy Sets and Fuzzy Logic. KR Chowdhary, Professor, Department of Computer Science & Engineering, MBM Engineering College, JNV University, Jodhpur,
Fuzzy Sets and Fuzzy Logic KR Chowdhary, Professor, Department of Computer Science & Engineering, MBM Engineering College, JNV University, Jodhpur, Outline traditional logic : {true,false} Crisp Logic
More informationLSN 4 Boolean Algebra & Logic Simplification. ECT 224 Digital Computer Fundamentals. Department of Engineering Technology
LSN 4 Boolean Algebra & Logic Simplification Department of Engineering Technology LSN 4 Key Terms Variable: a symbol used to represent a logic quantity Compliment: the inverse of a variable Literal: a
More informationANFIS: ADAPTIVE-NETWORK-BASED FUZZY INFERENCE SYSTEMS (J.S.R. Jang 1993,1995) bell x; a, b, c = 1 a
ANFIS: ADAPTIVE-NETWORK-ASED FUZZ INFERENCE SSTEMS (J.S.R. Jang 993,995) Membership Functions triangular triangle( ; a, a b, c c) ma min = b a, c b, 0, trapezoidal trapezoid( ; a, b, a c, d d) ma min =
More informationUsing Ones Assignment Method and. Robust s Ranking Technique
Applied Mathematical Sciences, Vol. 7, 2013, no. 113, 5607-5619 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2013.37381 Method for Solving Fuzzy Assignment Problem Using Ones Assignment
More informationGate Level Minimization Map Method
Gate Level Minimization Map Method Complexity of hardware implementation is directly related to the complexity of the algebraic expression Truth table representation of a function is unique Algebraically
More informationCaltech Harvey Mudd Mathematics Competition March 3, 2012
Team Round Caltech Harvey Mudd Mathematics Competition March 3, 2012 1. Let a, b, c be positive integers. Suppose that (a + b)(a + c) = 77 and (a + b)(b + c) = 56. Find (a + c)(b + c). Solution: The answer
More informationFuzzy Systems Handbook
The Fuzzy Systems Handbook Second Edition Te^hnische Universitat to instmjnik AutomatisiaMngstechnlk Fachgebi^KQegelup^stheorie und D-S4283 Darrftstadt lnvfentar-ngxc? V 2^s TU Darmstadt FB ETiT 05C Figures
More information