Slides:



Advertisements
Similar presentations
Fuzzy Set and Opertion. Outline Fuzzy Set and Crisp Set Expanding concepts Standard operation of fuzzy set Fuzzy relations Operations on fuzzy relations.
Advertisements

Basic Properties of Relations
Chapter 4 Systems of Linear Equations; Matrices Section 6 Matrix Equations and Systems of Linear Equations.
Fuzzy Expert Systems. Lecture Outline What is fuzzy thinking? What is fuzzy thinking? Fuzzy sets Fuzzy sets Linguistic variables and hedges Linguistic.
PART 2 Fuzzy sets vs crisp sets
Discrete Structures Chapter 5 Relations and Functions Nurul Amelina Nasharuddin Multimedia Department.
Lecture 3 Set Operations & Set Functions. Recap Set: unordered collection of objects Equal sets have the same elements Subset: elements in A are also.
Exercise 5.
CLASSICAL RELATIONS AND FUZZY RELATIONS
Theory and Applications
Extension Principle Adriano Cruz ©2002 NCE e IM/UFRJ
Ming-Feng Yeh General Fuzzy Systems A fuzzy system is a static nonlinear mapping between its inputs and outputs (i.e., it is not a dynamic system).
Let A = {1,2,3,4} and B = {a,b,c}. Define the relation f from A to B by f = {(1,b), (2,a), (3,c), (4,b)}. Is f a function? (1) Yes (2) No.
Fussy Set Theory Definition A fuzzy subset A of a universe of discourse U is characterized by a membership function which associate with each element.
1 Section 1.8 Functions. 2 Loose Definition Mapping of each element of one set onto some element of another set –each element of 1st set must map to something,
1 Section 7.1 Relations and their properties. 2 Binary relation A binary relation is a set of ordered pairs that expresses a relationship between elements.
Chapter 2 Fuzzy Sets Versus Crisp Sets
Classical Relations and Fuzzy Relations
Rule-Based Fuzzy Model. In rule-based fuzzy systems, the relationships between variables are represented by means of fuzzy if–then rules of the following.
RMIT University; Taylor's College
Functions.
Math 3121 Abstract Algebra I Lecture 3 Sections 2-4: Binary Operations, Definition of Group.
931102fuzzy set theory chap07.ppt1 Fuzzy relations Fuzzy sets defined on universal sets which are Cartesian products.
Chap. 6 Linear Transformations
3. Rough set extensions  In the rough set literature, several extensions have been developed that attempt to handle better the uncertainty present in.
Discrete Mathematics and Its Applications Sixth Edition By Kenneth Rosen Copyright  The McGraw-Hill Companies, Inc. Permission required for reproduction.
Section 4.2 Linear Transformations from R n to R m.
Functions1 Elementary Discrete Mathematics Jim Skon.
Sets Define sets in 2 ways  Enumeration  Set comprehension (predicate on membership), e.g., {n | n  N   k  k  N  n = 10  k  0  n  50} the set.
VENN DIAGRAM INDEX 1. DEFINATION OF VENN DIAGRAM. 2. VENN DIAGRAM OF A U B. 3. VENN DIAGRAM OF A  B. 4. VENN DIAGRAM OF A – B (ONLY A). 5. VENN DIAGRAM.
Chapter 5 – Relations and Functions. 5.1Cartesian Products and Relations Definition 5.1: For sets A, B  U, the Cartesian product, or cross product, of.
Practical Considerations When t , the supp, then there will be a value of t when supp folds, it becomes multi-w-to-one-x mapping.
Discrete Mathematics Relation.
 R  A×B,R is a relation from A to B , DomR  A 。  (a,b)  R (a, c)  R  (a,b)  R (a, c)  R unless b=c  function  DomR=A , (everywhere)function.
11 DISCRETE STRUCTURES DISCRETE STRUCTURES UNIT 5 SSK3003 DR. ALI MAMAT 1.
CHAPTER 3 FUZZY RELATION and COMPOSITION. 3.1 Crisp relation Product set Definition (Product set) Let A and B be two non-empty sets, the product.
Topic 2 Fuzzy Logic Control. Ming-Feng Yeh2-2 Outlines Basic concepts of fuzzy set theory Fuzzy relations Fuzzy logic control General Fuzzy System R.R.
Chap 3: Fuzzy Rules and Fuzzy Reasoning J.-S. Roger Jang ( 張智星 ) CS Dept., Tsing Hua Univ., Taiwan Fuzzy.
Ch.3 The Derivative. Differentiation Given a curve y = f(x ) Want to compute the slope of tangent at some value x=a. Let A=(a, f(a )) and B=(a +  x,
4. Relations and Digraphs Binary Relation Geometric and Algebraic Representation Method Properties Equivalence Relations Operations.
CLASSICAL RELATIONS AND FUZZY RELATIONS
Fuzzy Expert System n Introduction n Fuzzy sets n Linguistic variables and hedges n Operations of fuzzy sets n Fuzzy rules n Summary.
A function is a rule f that associates with each element in a set A one and only one element in a set B. If f associates the element b with the element.
1 Lecture 4 The Fuzzy Controller design. 2 By a fuzzy logic controller (FLC) we mean a control law that is described by a knowledge-based system consisting.
Binary Relations Definition: A binary relation R from a set A to a set B is a subset R ⊆ A × B. Example: Let A = { 0, 1,2 } and B = {a,b} {( 0, a), (
Functions – Page 1CSCI 1900 – Discrete Structures CSCI 1900 Discrete Structures Functions Reading: Kolman, Section 5.1.
Discrete Mathematics 3. MATRICES, RELATIONS, AND FUNCTIONS Lecture 6 Dr.-Ing. Erwin Sitompul
2004/10/5fuzzy set theory chap03.ppt1 Classical Set Theory.
Ch.3 Fuzzy Rules and Fuzzy Reasoning
Fuzzy Relations( 關係 ), Fuzzy Graphs( 圖 形 ), and Fuzzy Arithmetic( 運算 ) Chapter 4.
“It is impossible to define every concept.” For example a “set” can not be defined. But Here are a list of things we shall simply assume about sets. A.
Chapter 4 Fuzzy Graph and Relation Graph and Fuzzy Graph Graph G  (V, E) V : Set of vertices(node or element) E : Set of edges An edge is pair.
Chapter 3: Fuzzy Rules & Fuzzy Reasoning Extension Principle & Fuzzy Relations (3.2) Fuzzy if-then Rules(3.3) Fuzzy Reasonning (3.4)
Functions Section 2.3.
Chapter 4 Systems of Linear Equations; Matrices
Classical Relations and Fuzzy Relations
Chapter 3 FUZZY RELATION AND COMPOSITION
Cartesian product Given two sets A, B we define their Cartesian product is the set of all the pairs whose first element is in A and second in B. Note that.
Fuzzy Control Electrical Engineering Islamic University of Gaza
Relation and function.
Introduction to Fuzzy Logic
Financial Informatics –IX: Fuzzy Sets
Lecture 31 Fuzzy Set Theory (3)
Section 17.1 Parameterized Curves
Relations and their Properties
Introduction to Relations and Functions
Ch 5 Functions Chapter 5: Functions
Applied Discrete Mathematics Week 3: Sets
REVISION Relation. REVISION Relation Introduction to Relations and Functions.
Introduction to Fuzzy Set Theory
Presentation transcript:

2. Fuzzy Relations Objectives: Crisp and Fuzzy Relations Projections and Cylindrical Extensions* Extension Principle Compositions of Fuzzy Relations Ming-Feng Yeh

Cartesian Product: crisp Let A and B be two crisp subsets in X and Y, respectively. The Cartesian product of A and B, denoted by AB, is defined by Let X={0, 1}, Y={a,b,c}. If A=X and B=Y, then AB={(0,a), (0,b), (0,c), (1,a), (1,b), (1,c)} BA={(a,0), (b,0), (c,0), (a,1), (b,1), (c,1)} Ming-Feng Yeh

Fuzzy Relationships A fuzzy relationship over the pair X, Y is defined as a fuzzy subset of the Cartesian product XY. If X={0, 1}, Y={a,b,c}, then A = {0.1/(0,a), 0.6/(0,b), 0.8/(0,c), 0.3/(1,a), 0.5/(1,b), 0.7/(1,c)} is a fuzzy relationship over the space XY. Ming-Feng Yeh

Cartesian Product: fuzzy Let A and B be fuzzy sets in X and Y, respectively. The Cartesian product of A and B, denoted by AB, is a fuzzy set in the product space XY with the membership function: Assume X={0,1} and Y={a,b,c} Let A=1.0/0 + 0.6/1, B=0.2/a + 0.5/b+ 0.8/c. Then AB is a fuzzy relationship over XY. Ming-Feng Yeh

Cylindrical Extension* Assume X and Y are two crisp sets and let A be a fuzzy subset of X. The cylindrical extension of A to XY, denoted by , is a fuzzy relationship on XY. Assume X={a,b,c} and Y={1,2}. Let A={1/a, 0.6/b, 0.3/c}. Then the cylindrical extension of A to XY is {1/(a,1), 1/(a,2), 0.6/(b,1), 0.6/(b,2), 0.3/(c,1), 0.3/(c,2)} Ming-Feng Yeh

Cylindrical Extension* A(x) x y x Ming-Feng Yeh

Projection* Assume A is a fuzzy relationship on XY. The projection of A onto X is a fuzzy subset A of X, denoted by A=Projx A, Assume X = {a,b,c} and Y = {1,2}. Let A={1/(a,1), 0.6/(a,2), 0.8/(b,1), 0.6/(b,2), 0.3/(c,1), 0.5/(c,2)}. Then Projx A = {1/a, 0.6/b, 0.5/c}. Projy A = {0.8/1, 0.6/2}. Ming-Feng Yeh

Projection* Ming-Feng Yeh

Extension Principle Assume X and Y are two crisp sets and let f be a mapping form X into Y, f: XY, such that xX, f(x) = y Y. Assume A is a fuzzy subset of X, using the extension principle, we can define f(A) as a fuzzy subset of Y such that Denote B = f(A), then B is a fuzzy subset of Y such that for each y Y Ming-Feng Yeh

Example 2-3 Assume X = {1, 2, 3} and Y = {a, b, c, d, e}. Let f be defined by f(1) = a, f(2) = e, f(3) = b. Let A = {1.0/1, 0.3/2, 0.7/3} be a fuzzy subset, then B = f(A) = {1.0/a, 0.3/e, 0.7/b}. Let A = 0.1/2 + 0.4/1 + 0.8/0 + 0.9/1 + 0.3/2 and f(x) = x2 3. Then B = 0.1/1 + 0.4/2 + 0.8/3 + 0.9/2 + 0.3/1 = 0.8/3 + (0.40.9)/2 + (0.10.3)/1 = 0.8/3 + 0.9/2 + 0.3/1 Ming-Feng Yeh

Binary Fuzzy Relations Let X and Y be two universes of discourse. Then is a binary fuzzy relation in XY. Examples of binary fuzzy relation: y is much greater than x. (x and y are numbers) x is close to y. (x and y are numbers) x depends on y. (x and y are events) x and y look alike. (x and y are persons, objects, etc.) If x is large, then y is small. (x is an observed reading and y is a corresponding action) Ming-Feng Yeh

Max-min Composition Let R1 and R2 be two fuzzy relations defined on XY and YZ, respectively. The max-min composition of R1 and R2 is a fuzzy set defined by Max-min product: the calculation of is almost the same as matrix multiplication, except that  and  are replaced by  and , respectively. Ming-Feng Yeh

Max-product Composition Let R1 and R2 be two fuzzy relations defined on XY and YZ, respectively. The max-product composition of R1 and R2 is a fuzzy set defined by Ming-Feng Yeh

Example 2-3 R1 = “x is relevant to y”, R2 = “y is relevant to z”, X = {1,2,3}, Y={,,,} and Z={a,b}. Max-min composition: Max-product composition: Ming-Feng Yeh