Summary. Operations on Fuzzy Sets. Zadeh s Definitions. Zadeh s Operations TNorms SNorms. Properties of Fuzzy Sets Fuzzy Measures


 Georgiana Briggs
 2 years ago
 Views:
Transcription
1 Summary Operations on Fuzzy Sets Zadeh s Operations TNorms SNorms Adriano Cruz 00 NCE e IM/UFRJ Properties of Fuzzy Sets Fuzzy Measures One should not increase, beyond what is necessary, the number of entities required to explain anything. Occam's Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets Zadeh s Definitions Lofty Zadeh put forward the basic set operations is his seminal paper Fuzzy Sets, Information and Control, 1965 These operations reduce to the boolean operations when crisp sets are Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets 3
2 Why these operators? The crisp set operators are very well defined and understood, however when fuzzy sets are considered this definition is fuzzy and many other operations can be considered. Fuzzy set operators must obey a set of rules that generalize the operations. The so called Tnorms (T(x,y)) and the Tconorms or Snorms (S(x,y)). Tnorms generalize the and operator and t conorms generalize the or operator T Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets 8 Intersection operation Tnorms Any tnorm operator, denoted as t(x,y) must satisfy five axioms. Tnorms map from [0,1]x[0,1] [0,1] Let µ A (x), µ B (x), µ C (x) and µ D (x) four functions (sets). In order to simplify the notation we will use the letters a, b, c e d to represent them. T.1 T(0,0) = 0 T. T(a,b) = T(b,a) commutative T.3 T(a,1) = a neuter T.4 T(T(a,b),c)=T(a,T(b,c)) associative T.5 T(c,d) <=T(a,b) if c<=a and d<=b Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets 10 Tnorms: comments Minimum, Tnorm? It can be proved that the minimum operation is a tnorm The product operator is also a tnorm Obviously there are other operations that satisfy these axioms It can be proved that for any tnorm Τ(µ Α (x), µ Β (x)) <= min(µ Α (x), µ Β (x)) T.1 min(0,0) = 0 T. min(a,b) = min(b,a) T.3 min(a,1) = a T.4 min(min(a,b),c) = min(a,min(b,c)) = min(a,b,c) T.5 min(c,d) <= min(a,b) if c <= a and d <= Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets 1
3 Union Any snorm operator denoted as s(x,y) must satisfy five axioms Snorms map [0,1]x[0,1] [0,1] Let µ A (x), µ B (x), µ C (x) e µ D (x) four fuzzy sets. In order to simplify the notation we will use the letters a,b,c and d. Snorms or Tconorms S.1 S(1,1) = 1 S. S(a,b) = S(b,a) commutative S.3 S(a,0) = a neuter S.4 S(S(a,b),c)=S(a,S(b,c)) associative S.5 S(c,d) <=S(a,b) if c<=a and d<=b Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets 14 Snorms: comments 1 It can be proved that the maximum is a s norm Obviously there are other operations that satisfy these axioms. The addition operation do not satisfy the S.1 axiom, so it can not be used. It can be proved that for any Snorm we have S(µ Α (x), µ Β (x)) >= max(µ Α (x), µ Β (x)) Snorms: comments Note that it is not required the Snorm to be idempotent, that is S(a,a)=a, therefore the union of a set to itself it is not required to be equal to itself. Nor is required that the Snorm to be Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets 16 Maximum, Snorm? Algebraic sum, Snorm? S.1 max(0,0) = 0 S. max(a,b) = max(b,a) S.3 max(a,1) = a S.4 max(max(a,b),c) = max(a,max(b,c)) = max(a,b,c) S.5 max(c,d) <= max(a,b) if c <= a and d <= b µ A B a,b =a+b a b S S. a+b a b=b+a b a comutativa S. 3 a+ 0 a 0 Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets 18
4 Algebraic Sum, Snorm? S.4 S((a+b),c) = (a+bab) + c (a+bab)c = a+bab+cacbc+abc = a+(b+cbc)a(b+cbc) = S(a,(b+c)) = a + b + c ab ac bc + abc S.5 if c <= a, d <= b, a, b, c, d <= 1 a + b ab >= c + d cd Other examples Prove that T(a,b) <= min(a,b) T5: T(a,b) <= T(a,1) = a T: T(a,b) = T(b,a) T5: T(b,a) <= T(b,1) = b T: T(a,b) <= Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets 0 Pairs of Tnorms and Snorms Tnorm  Drastic Product: min x,y if max x,y 1 DP x,y 0 x,y 1 Snorm  Drastic Sum: max x,y if min x,y 0 DS x,y 1 x,y> 0 Pairs of Tnorms and Snorms Tnorm  Bounded Difference: BD x,y max 0, x+y 1 Snorm  Bounded Sum: BS x,y min 1, Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets Pairs of Tnorms and Snorms Tnorm Einstein Product: Pairs of Tnorms and Snorms Tnorm Algebraic Product: xy EP x,y x+y xy Snorm  Einstein Sum: ES x,y x+y 1+x. y AP x,y =xy Snorm  Algebraic Sum: AS x,y =x+y Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets 4
5 Pairs of Tnorms and Snorms Four Tnorm operators Tnorm Hamacher Product: HP x,y xy x+y xy Snorm  Hamacher Sum: x+y xy HS x,y 1 Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets 6 Four Tconorm operators Pairs of Tnorms and Snorms Tnorm DuboisPrade: xy DPr T x,y max p,x,y Obs. p is a parameter that ranges from 0 to 1. Snorm DuboisPrade: x+y xy min 1 p,x,y DPr S x,y max p,1 x,1 Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets 8 DuboisPrade Operators When p=1 DuboisPrade Tnorm becomes the Algebraic Product (xy) DuboisPrade Snorm becomes the Algebraic Sum (x+yxy) When p=0 DuboisPrade Tnorm becomes the min(xy) DuboisPrade Snorm becomes the max(xy) Pairs of Tnorms and Snorms Tnorm Yager: Y T x,y 1 min 1, 1 x p 1 y p 1 p Obs. p is a parameter that ranges from 0 to. Snorm Yager: Y T x,y min 1, x p +y p 1 Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets 30
6 Yager Operators When p=1.0 Yager Tnorm becomes the bounded difference (max(0,x+y1)) Yager Snorm becomes the bounded sum (min(1,x+y)) Complement When p> Yager Tnorm converges to min(x,y) Yager Snorm converges to Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets 3 Fuzzy Complement axioms Fuzzy Complements A fuzzy complement operator is a continuous function N:[0,1] :[0,1] [0,1] which meets the following axioms: N(0)=1 and N(1) = 0 (boundary) (a.1) N(a) N(b) ) if a b (monotonicity) (a.) Another optional requirements are N(x) is continuous (a.3) N(N(a))= ))=a (involution) (a.4) Continuous a.3 All functions a.1 e a. Involutive a.4 Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets 34 Usual Fuzzy Complement Consider  N(x)=1x N(0)=1 and N(1) = 0 (boundary) N(a) N(b) if a b N(N(x))=1(1x)= )=x Continuous in the interval (monotonicity) Another Ex of Complement 1 for a t Consider N x 0 a>t 1 Example of complement 0.9 t,a 0,1 N (x) Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets 36
7 Another Ex of Complement 1 Consider N x 1 for a t 0 a>t t,a 0,1 Satisfy only the axiomatic requirements: N(1)=0, N(0)=1 N(x) ) is monotonic N(x) ) is not continuous N(x) ) is not involutive Sugeno s complement The operator is defined as N s a 1 a 1+sa where s is a parameter greater than 1. For each s, we obtain a particular Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets 38 Sugeno s complement Yager s complement The operator is defined as N y a 1 a y 1 y where y is a positive Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets 40 Yager s complement Complement equilibrium The point of equilibrium is any x for which N(x)=x For a classical fuzzy set x=1x x = 0.5 Equilibrium can be used to measure Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets 4
8 Fuzzy Sets Properties Properties Comutativity Associativity Distributivity Absorption A B=B A A B=B A A B C A B C A B C A B C A B C A B A C A B C A B A C A A B =A A A B Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets 44 Fuzzy Sets Properties Identity De Morgan A =A A A X=X A X=A A B A B A B A Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets 45 Checking Properties Remember that a if a b min a,b = b if b<a a+b a b min a,b = and b if a b max a,b = a if b<a a+b+ a b max a,b Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets 46 Checking Properties Checking Properties Lets check the absorption property A A B =A max µ A x,min µ A x,µ B x =µ A x A B = A A B = a+b a b a+ a+b a b a a+b a b A A B = = If a b A A B = A A B = 3a+b a b a b+ a b 3a +b a b a b+ a b 4 3a +b a b a b a b 4 Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets 48
9 Checking Properties Laws of Aristotle A A B = if a<b A A B = A A B = 3a +b a b a b+ a b 4 3a +b+ a b a b a b 4 a Law of NonContradiction: One cannot say of something that it is and that it is not in the same respect and at the same time. One element must belong to a set or its complement. Since the intersection between one set and its complement may be not empty we may have A Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets 50 Law of noncontradiction Laws of Aristotle 1.0 Non adults adults Law of excluded middle: for any proposition P, it is true that (P or notp). adults adults So the union of a set and its complement should give all the universe adults adults However the result may be not the universe A A Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets 5 Law of noncontradiction 1.0 Non adults adults Measuring adults adults Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets 54
10 Fuzzy Entropy Fuzzy Entropy The entropy of a fuzzy set is defined as c A A E A c A A c is a counting operation (addition or integration) defined over the set. Note that for a crisp set the numerator is always 0 and the entropy of a crisp set is always Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets 55 1 The entropy of the adult fuzzy set is c A A = 5 c A A = = 35 E A = 5 35 = 0.14 No adults adults adult Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets 56 Fuzziness Measurements A measure of fuzziness is a function f : X P(X) is the set of all fuzzy subsets of X There are three requirements that a meaningful measure must satisfy Only one is unique; the other depend on the meaning of fuzziness Requirements F1: f(a) = 0 iff A is a crisp set A B means A is less fuzzy than B F: if A B then f A f B F3: f(a) assumes the maximum value iff A is maximally Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets 58 Measurement Based on Distance One measure of fuzziness is defined in terms of a metric distance from the set A to the nearest crisp set. Distance from point A(a 0,a 1,,a n ) to B(b 0,b 1,,b n ) d p A,B p n i=1 a i b i p Measurement Based on Distance If p= the distance is the Euclidean distance, if p=1 the distance it is the Hamming distance. d p A,B p n i=1 a i b i Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets 60
11 The Geometry of Sets Crisp Sets can be view as points in a space Fuzzy sets are also part of the same space Using these concepts it is possible to measure distances from crisp to fuzzy Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets 61 Classic Power Set Classic Power Set: the set of all subsets of a classic set. Let consider X={x 1,x,x 3 } Power Set is represented by X X ={, {x 1 }, {x }, {x 3 }, {x 1,x }, {x 1,x 3 }, {x,x 3 }, Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets 6 Vertices The vertices in space The 8 sets can correspond to 8 vectors (0,0,1) x3 (1,0,1) X ={, {x 1 }, {x }, {x 3 }, {x 1,x }, {x 1,x 3 }, {x,x 3 }, X} (0,1,1) (1,1,1) X ={(0,0,0),(1,0,0),(0,1,0),(0,0,1),(1,1,0), (1,0,1),(0,1,1),(1,1,1)} The 8 sets are the vertices of a Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets 63 (0,1,0) x x (0,0,0) 1 (1,0,0) Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets 64 Fuzzy Power Set The Fuzzy Cube The Fuzzy Power set is the set of all fuzzy (0,0,1) x 3 (1,0,1) subsets of X={x 1,x,x 3 } It is represented by F( X ) A Fuzzy subset of X is a point in a cube (0,1,1) 0.7 (1,1,1) A={(x 1,0.5),(x,0.3),(x 3,0.7)} The Fuzzy Power set is the unit hypercube 0.3 x (0,1,0) 0.5 (1,1,0) x1 Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets 66
12 Fuzzy Distances Let X={x 1,x } and A={(x 1,1/3),(x,3/4)} Maximal Fuzziness x Let X={x 1 1,x } and A={(x 1,1/),(x,1/)} x 1 n=1 3/4 n= A={(x1,1/3), (x,3/4)} /4 A={(x1,1/), (x,1/)} /4 n= n=1 1/4 0,0 1/ 1 x1 0,0 1/3 /3 1 Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets 68 Fuzzy Operations Fuzzy Operations in the Space Let X={x 1,x } and A={(x 1,1/3),(x,3/4)} Let A represent the complement of A A ={(x 1,/3),(x,1/4)} (0,1) 3/4 x A(1/3,3/4) A A (1,1) A A ={(x 1,/3),(x,3/4)} A A ={(x 1,1/3),(x,1/4)} 1/4 φ A A 1/3 /3 A (/3,1/4) (1,0) x Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets 70 Paradox at the Midpoint Classical logic forbids the middle point by the noncontradiction and excluded middle axioms The Liar from Crete Let S be he is a liar, let nots be he is not a liar Since S nots and nots S t(s)=t(nots)=1t(s) t(s)=0.5 Cardinality of a Fuzzy Set The cardinality of a fuzzy set is equal to the sum of the membership degrees of all elements. The cardinality is represented by A n A µ A x i Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets 7
13 Distance The distance d p between two sets represented by points in the space is defined as d p A,B p n i=1 µ A x i µ B x i p If p= the distance is the Euclidean distance, if p=1 the distance it is the Hamming distance Distance and Cardinality If the point B is the empty set (the origin) d 1 A,O µ A x i 0 i=1 So the cardinality of a fuzzy set is the Hamming distance to the origin n d 1 A,O A µ A x i n i= Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets 74 Fuzzy Cardinality Fuzzy Entropy (0,1) x (1,1) How fuzzy is a fuzzy set? 3/4 A Fuzzy entropy varies from 0 to 1. Cube vertices has entropy 0. A =d 1 (A,φ) The middle point has entropy 1. φ 1/3 (1,0) x Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets 76 Fuzzy Operations in the Space Fuzzy Entropy Geometry (0,1) 3/4 x A A A (1,1) (0,1) 3/4 x a A (1,1) 1/4 A A A b φ (1,0) 1/3 /3 x1 A A E A A Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets 77 φ 1/3 E A a b d1 A,A near d 1 A,A far Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets 78 x1
14 Fuzzy entropy, max and min T(x,y) min(x,y) max(x,y) S(x,y) So the value of 1 for the middle point does not hold when other Tnorm is chosen. Let A= {(x 1,0.5),(x,0.5)} E(A)=0.5/0.5=1 Let T(x,y)=x.y and C(x,y)=x+yxy E(A)=0.5/0.75=0.333 Subsets Sets contain subsets. A is a subset of B (A B) iff every element of A is an element of B. A is a subset of B iff A belongs to the power set of B (A B iff A B Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets 80 Subsethood examples Consider A={(x 1,1/3),(x =1/)} and B={(x 1,1/),(x =3/4)} A B, but B A (0,1) 3/4 1/ x A B (1,1) Not Fuzzy Subsethood The so called membership dominated definition is not fuzzy. The fuzzy power set of B (F( B )) is the hyper rectangle docked at the origin of the hyper cube. Any set is either a subset or not. φ 1/3 1/ (1,0) Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets 8 Fuzzy power set size F( B ) has infinity cardinality. For finite dimensional sets the size of F( B ) is the Lebesgue measure or x volume V(B) (1,1) (0,1) n V B µ B x i i=1 3/4 1/ A B Fuzzy Subsethood Let S(A,B)=Degree(A B)=µ F( B ) (A) Suppose only element j violates µ A (x j ) µ B (x j ), so A is not totally subset of B. Counting violations and their magnitudes shows the degree of subsethood. (1,0) φ 1/3 1/ x Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets 84
15 Fuzzy Subsethood Supersethood(A,B)=1S(A,B) Sum all violations=max(0,µ A (x j )µ B (x j )) 0 S(A,B) 1 x X max 0, µ A x µ B x Supersethood A,B A max 0, µ A x µ B x S A,B 1 x X A Subsethood measures Consider A={(x 1,0.5),(x =0.5)} and B={(x 1,0.5),(x =0.9)} max 0, max 0, S A,B = S B,A = 1 S A,B max 0, max 0, S B,A = Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets Adriano Cruz NCE e IM  UFRJ Operations of Fuzzy Sets 86
Computational Intelligence
Plan for Today Computational Intelligence Winter Term 2/2 Fuzzy sets Axioms of fuzzy complement, t and snorms Generators Dual tripels Prof. Dr. Günter Rudolph Lehrstuhl für Algorithm Engineering (LS
More informationModels for Inexact Reasoning. Fuzzy Logic Lesson 1 Crisp and Fuzzy Sets. Master in Computational Logic Department of Artificial Intelligence
Models for Inexact Reasoning Fuzzy Logic Lesson 1 Crisp and Fuzzy Sets Master in Computational Logic Department of Artificial Intelligence Origins and Evolution of Fuzzy Logic Origin: Fuzzy Sets Theory
More informationFuzzy sets I. Prof. Dr. Jaroslav Ramík
Fuzzy sets I Prof. Dr. Jaroslav Ramík Fuzzy sets I Content Basic definitions Examples Operations with fuzzy sets (FS) tnorms and tconorms Aggregation operators Extended operations with FS Fuzzy numbers:
More informationSome Definitions about Sets
Some Definitions about Sets Definition: Two sets are equal if they contain the same elements. I.e., sets A and B are equal if x[x A x B]. Notation: A = B. Recall: Sets are unordered and we do not distinguish
More informationA set is a Many that allows itself to be thought of as a One. (Georg Cantor)
Chapter 4 Set Theory A set is a Many that allows itself to be thought of as a One. (Georg Cantor) In the previous chapters, we have often encountered sets, for example, prime numbers form a set, domains
More informationThe Language of Mathematics
CHPTER 2 The Language of Mathematics 2.1. Set Theory 2.1.1. Sets. set is a collection of objects, called elements of the set. set can be represented by listing its elements between braces: = {1, 2, 3,
More informationSections 2.1, 2.2 and 2.4
SETS Sections 2.1, 2.2 and 2.4 Chapter Summary Sets The Language of Sets Set Operations Set Identities Introduction Sets are one of the basic building blocks for the types of objects considered in discrete
More informationCHAPTER 2 OPERATIONS ON FUZZY SETS. Remark: 1. All functions that satisfy axioms c1 and c2 form the most general class of fuzzy complements.
CHPTER OPERTIONS ON FUZZY SETS Fuzzy Complement Fuzzy Union Fuzzy Intersection Combinations of Operations General ggregation Operations Fuzzy Complement Remark: ll functions that satisfy axioms c and c
More informationClassical Sets and Fuzzy Sets Classical Sets Operation on Classical Sets Properties of Classical (Crisp) Sets Mapping of Classical Sets to Functions
Classical Sets and Fuzzy Sets Classical Sets Operation on Classical Sets Properties of Classical (Crisp) Sets Mapping of Classical Sets to Functions Fuzzy Sets Notation Convention for Fuzzy Sets Fuzzy
More informationBoolean Algebra Part 1
Boolean Algebra Part 1 Page 1 Boolean Algebra Objectives Understand Basic Boolean Algebra Relate Boolean Algebra to Logic Networks Prove Laws using Truth Tables Understand and Use First Basic Theorems
More informationAnnouncements. CompSci 230 Discrete Math for Computer Science Sets. Introduction to Sets. Sets
CompSci 230 Discrete Math for Computer Science Sets September 12, 2013 Prof. Rodger Slides modified from Rosen 1 nnouncements Read for next time Chap. 2.32.6 Homework 2 due Tuesday Recitation 3 on Friday
More informationCourse 221: Analysis Academic year , First Semester
Course 221: Analysis Academic year 200708, First Semester David R. Wilkins Copyright c David R. Wilkins 1989 2007 Contents 1 Basic Theorems of Real Analysis 1 1.1 The Least Upper Bound Principle................
More informationFoundations of Geometry 1: Points, Lines, Segments, Angles
Chapter 3 Foundations of Geometry 1: Points, Lines, Segments, Angles 3.1 An Introduction to Proof Syllogism: The abstract form is: 1. All A is B. 2. X is A 3. X is B Example: Let s think about an example.
More informationFS I: Fuzzy Sets and Fuzzy Logic. L.A.Zadeh, Fuzzy Sets, Information and Control, 8(1965)
FS I: Fuzzy Sets and Fuzzy Logic Fuzzy sets were introduced by Zadeh in 1965 to represent/manipulate data and information possessing nonstatistical uncertainties. L.A.Zadeh, Fuzzy Sets, Information and
More informationSection 8.8. 1. The given line has equations. x = 3 + t(13 3) = 3 + 10t, y = 2 + t(3 + 2) = 2 + 5t, z = 7 + t( 8 7) = 7 15t.
. The given line has equations Section 8.8 x + t( ) + 0t, y + t( + ) + t, z 7 + t( 8 7) 7 t. The line meets the plane y 0 in the point (x, 0, z), where 0 + t, or t /. The corresponding values for x and
More informationMATH REVIEW KIT. Reproduced with permission of the Certified General Accountant Association of Canada.
MATH REVIEW KIT Reproduced with permission of the Certified General Accountant Association of Canada. Copyright 00 by the Certified General Accountant Association of Canada and the UBC Real Estate Division.
More informationGeometry Unit 1. Basics of Geometry
Geometry Unit 1 Basics of Geometry Using inductive reasoning  Looking for patterns and making conjectures is part of a process called inductive reasoning Conjecture an unproven statement that is based
More informationSets. A set is a collection of (mathematical) objects, with the collection treated as a single mathematical object.
Sets 1 Sets Informally: A set is a collection of (mathematical) objects, with the collection treated as a single mathematical object. Examples: real numbers, complex numbers, C integers, All students in
More informationChapter 4 BOOLEAN ALGEBRA AND THEOREMS, MIN TERMS AND MAX TERMS
Chapter 4 BOOLEAN ALGEBRA AND THEOREMS, MIN TERMS AND MAX TERMS Lesson 5 BOOLEAN EXPRESSION, TRUTH TABLE and product of the sums (POSs) [MAXTERMS] 2 Outline POS two variables cases POS for three variable
More informationNOTES ON MEASURE THEORY. M. Papadimitrakis Department of Mathematics University of Crete. Autumn of 2004
NOTES ON MEASURE THEORY M. Papadimitrakis Department of Mathematics University of Crete Autumn of 2004 2 Contents 1 σalgebras 7 1.1 σalgebras............................... 7 1.2 Generated σalgebras.........................
More informationThe set consisting of all natural numbers that are in A and are in B is the set f1; 3; 5g;
Chapter 5 Set Theory 5.1 Sets and Operations on Sets Preview Activity 1 (Set Operations) Before beginning this section, it would be a good idea to review sets and set notation, including the roster method
More information35 Multicriteria evaluation and GIS
35 Multicriteria evaluation and GIS J R EASTMAN Multicriteria evaluation in GIS is concerned with the allocation of land to suit a specific objective on the basis of a variety of attributes that the
More informationA Foundation for Geometry
MODULE 4 A Foundation for Geometry There is no royal road to geometry Euclid 1. Points and Lines We are ready (finally!) to talk about geometry. Our first task is to say something about points and lines.
More informationCM2202: Scientific Computing and Multimedia Applications General Maths: 2. Algebra  Factorisation
CM2202: Scientific Computing and Multimedia Applications General Maths: 2. Algebra  Factorisation Prof. David Marshall School of Computer Science & Informatics Factorisation Factorisation is a way of
More informationBEGINNING ALGEBRA ACKNOWLEDMENTS
BEGINNING ALGEBRA The Nursing Department of Labouré College requested the Department of Academic Planning and Support Services to help with mathematics preparatory materials for its Bachelor of Science
More informationClassical Analysis I
Classical Analysis I 1 Sets, relations, functions A set is considered to be a collection of objects. The objects of a set A are called elements of A. If x is an element of a set A, we write x A, and if
More informationMathematics Review for MS Finance Students
Mathematics Review for MS Finance Students Anthony M. Marino Department of Finance and Business Economics Marshall School of Business Lecture 1: Introductory Material Sets The Real Number System Functions,
More informationBasic Properties of Rings
Basic Properties of Rings A ring is an algebraic structure with an addition operation and a multiplication operation. These operations are required to satisfy many (but not all!) familiar properties. Some
More informationC relative to O being abc,, respectively, then b a c.
2 EPProgram  Strisuksa School  Roiet Math : Vectors Dr.Wattana Toutip  Department of Mathematics Khon Kaen University 200 :Wattana Toutip wattou@kku.ac.th http://home.kku.ac.th/wattou 2. Vectors A
More informationCardinality of Fuzzy Sets: An Overview
Cardinality of Fuzzy Sets: An Overview Mamoni Dhar Asst. Professor, Department Of Mathematics Science College, Kokrajhar783370, Assam, India mamonidhar@rediffmail.com, mamonidhar@gmail.com Abstract In
More informationConventional Sets and Fuzzy Sets
Conventional Sets and Conventional Sets A set is a collection of things, for example the room temperature, the set of all real numbers, etc. 2 Conventional Sets Such collection of things are called the
More informationAnalysis MA131. University of Warwick. Term
Analysis MA131 University of Warwick Term 1 01 13 September 8, 01 Contents 1 Inequalities 5 1.1 What are Inequalities?........................ 5 1. Using Graphs............................. 6 1.3 Case
More informationA matrix over a field F is a rectangular array of elements from F. The symbol
Chapter MATRICES Matrix arithmetic A matrix over a field F is a rectangular array of elements from F The symbol M m n (F) denotes the collection of all m n matrices over F Matrices will usually be denoted
More informationSection 6.1 Factoring Expressions
Section 6.1 Factoring Expressions The first method we will discuss, in solving polynomial equations, is the method of FACTORING. Before we jump into this process, you need to have some concept of what
More informationCopy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any.
Algebra 2  Chapter Prerequisites Vocabulary Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. P1 p. 1 1. counting(natural) numbers  {1,2,3,4,...}
More informationAutomata Theory. Şubat 2006 Tuğrul Yılmaz Ankara Üniversitesi
Automata Theory Automata theory is the study of abstract computing devices. A. M. Turing studied an abstract machine that had all the capabilities of today s computers. Turing s goal was to describe the
More informationSo let us begin our quest to find the holy grail of real analysis.
1 Section 5.2 The Complete Ordered Field: Purpose of Section We present an axiomatic description of the real numbers as a complete ordered field. The axioms which describe the arithmetic of the real numbers
More informationTHREE DIMENSIONAL GEOMETRY
Chapter 8 THREE DIMENSIONAL GEOMETRY 8.1 Introduction In this chapter we present a vector algebra approach to three dimensional geometry. The aim is to present standard properties of lines and planes,
More informationINTRODUCTORY SET THEORY
M.Sc. program in mathematics INTRODUCTORY SET THEORY Katalin Károlyi Department of Applied Analysis, Eötvös Loránd University H1088 Budapest, Múzeum krt. 68. CONTENTS 1. SETS Set, equal sets, subset,
More informationMATH 304 Linear Algebra Lecture 24: Scalar product.
MATH 304 Linear Algebra Lecture 24: Scalar product. Vectors: geometric approach B A B A A vector is represented by a directed segment. Directed segment is drawn as an arrow. Different arrows represent
More informationGeometric Transformations
Geometric Transformations Definitions Def: f is a mapping (function) of a set A into a set B if for every element a of A there exists a unique element b of B that is paired with a; this pairing is denoted
More informationPART I. THE REAL NUMBERS
PART I. THE REAL NUMBERS This material assumes that you are already familiar with the real number system and the representation of the real numbers as points on the real line. I.1. THE NATURAL NUMBERS
More informationChapter 1. Logic and Proof
Chapter 1. Logic and Proof 1.1 Remark: A little over 100 years ago, it was found that some mathematical proofs contained paradoxes, and these paradoxes could be used to prove statements that were known
More informationSets and Logic. Chapter Sets
Chapter 2 Sets and Logic This chapter introduces sets. In it we study the structure on subsets of a set, operations on subsets, the relations of inclusion and equality on sets, and the close connection
More informationVector and Matrix Norms
Chapter 1 Vector and Matrix Norms 11 Vector Spaces Let F be a field (such as the real numbers, R, or complex numbers, C) with elements called scalars A Vector Space, V, over the field F is a nonempty
More informationThis chapter describes set theory, a mathematical theory that underlies all of modern mathematics.
Appendix A Set Theory This chapter describes set theory, a mathematical theory that underlies all of modern mathematics. A.1 Basic Definitions Definition A.1.1. A set is an unordered collection of elements.
More informationexpression is written horizontally. The Last terms ((2)( 4)) because they are the last terms of the two polynomials. This is called the FOIL method.
A polynomial of degree n (in one variable, with real coefficients) is an expression of the form: a n x n + a n 1 x n 1 + a n 2 x n 2 + + a 2 x 2 + a 1 x + a 0 where a n, a n 1, a n 2, a 2, a 1, a 0 are
More information16. Let S denote the set of positive integers 18. Thus S = {1, 2,..., 18}. How many subsets of S have sum greater than 85? (You may use symbols such
æ. Simplify 2 + 3 + 4. 2. A quart of liquid contains 0% alcohol, and another 3quart bottle full of liquid contains 30% alcohol. They are mixed together. What is the percentage of alcohol in the mixture?
More informationCS 341 Homework 9 Languages That Are and Are Not Regular
CS 341 Homework 9 Languages That Are and Are Not Regular 1. Show that the following are not regular. (a) L = {ww R : w {a, b}*} (b) L = {ww : w {a, b}*} (c) L = {ww' : w {a, b}*}, where w' stands for w
More information1.3 Polynomials and Factoring
1.3 Polynomials and Factoring Polynomials Constant: a number, such as 5 or 27 Variable: a letter or symbol that represents a value. Term: a constant, variable, or the product or a constant and variable.
More informationThe Straight Line I. INTRODUCTORY
UNIT 1 The Straight Line I. INTRODUCTORY Geometry is the science of space and deals with the shapes, sizes and positions of things. Euclid was a Greek mathematician of the third century B.C. who wrote
More informationCHAPTER 2. Set, Whole Numbers, and Numeration
CHAPTER 2 Set, Whole Numbers, and Numeration 2.1. Sets as a Basis for Whole Numbers A set is a collection of objects, called the elements or members of the set. Three common ways to define sets: (1) A
More information1.4. Arithmetic of Algebraic Fractions. Introduction. Prerequisites. Learning Outcomes
Arithmetic of Algebraic Fractions 1.4 Introduction Just as one whole number divided by another is called a numerical fraction, so one algebraic expression divided by another is known as an algebraic fraction.
More informationCOMPRESSION SPRINGS: STANDARD SERIES (INCH)
: STANDARD SERIES (INCH) LC 014A 01 0.250 6.35 11.25 0.200 0.088 2.24 F F M LC 014A 02 0.313 7.94 8.90 0.159 0.105 2.67 F F M LC 014A 03 0.375 9.52 7.10 0.126 0.122 3.10 F F M LC 014A 04 0.438 11.11 6.00
More informationIncenter Circumcenter
TRIANGLE: Centers: Incenter Incenter is the center of the inscribed circle (incircle) of the triangle, it is the point of intersection of the angle bisectors of the triangle. The radius of incircle is
More informationBaltic Way 1995. Västerås (Sweden), November 12, 1995. Problems and solutions
Baltic Way 995 Västerås (Sweden), November, 995 Problems and solutions. Find all triples (x, y, z) of positive integers satisfying the system of equations { x = (y + z) x 6 = y 6 + z 6 + 3(y + z ). Solution.
More informationLecture 4: Random Variables
Lecture 4: Random Variables 1. Definition of Random variables 1.1 Measurable functions and random variables 1.2 Reduction of the measurability condition 1.3 Transformation of random variables 1.4 σalgebra
More informationDigital Logic Design
Digital Logic Design ENGG1015 1 st Semester, 2010 Dr. Kenneth Wong Dr. Hayden So Department of Electrical and Electronic Engineering Determining output level from a diagram Implementing Circuits From Boolean
More informationImprecise probabilities, bets and functional analytic methods in Łukasiewicz logic.
Imprecise probabilities, bets and functional analytic methods in Łukasiewicz logic. Martina Fedel joint work with K.Keimel,F.Montagna,W.Roth Martina Fedel (UNISI) 1 / 32 Goal The goal of this talk is to
More informationGeometry in a Nutshell
Geometry in a Nutshell Henry Liu, 26 November 2007 This short handout is a list of some of the very basic ideas and results in pure geometry. Draw your own diagrams with a pencil, ruler and compass where
More informationChapter 1. SigmaAlgebras. 1.1 Definition
Chapter 1 SigmaAlgebras 1.1 Definition Consider a set X. A σ algebra F of subsets of X is a collection F of subsets of X satisfying the following conditions: (a) F (b) if B F then its complement B c is
More information3. INNER PRODUCT SPACES
. INNER PRODUCT SPACES.. Definition So far we have studied abstract vector spaces. These are a generalisation of the geometric spaces R and R. But these have more structure than just that of a vector space.
More information4 BOOLEAN ALGEBRA AND LOGIC SIMPLIFICATION
4 BOOLEAN ALGEBRA AND LOGIC SIMPLIFICATION 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 informationAnother fuzzy membership function that is often used to represent vague, linguistic terms is the Gaussian which is given by:
Gaussian Membership Functions Another fuzzy membership function that is often used to represent vague, linguistic terms is the Gaussian which is given by: µ A i(x) = exp( (c i x) 2 ), (1) 2σ 2 i where
More informationDiscrete Mathematics Set Operations
Discrete Mathematics 13. Set Operations Introduction to Set Theory A setis a new type of structure, representing an unordered collection (group, plurality) of zero or more distinct (different) objects.
More informationIntroduction Russell s Paradox Basic Set Theory Operations on Sets. 6. Sets. Terence Sim
6. Sets Terence Sim 6.1. Introduction A set is a Many that allows itself to be thought of as a One. Georg Cantor Reading Section 6.1 6.3 of Epp. Section 3.1 3.4 of Campbell. Familiar concepts Sets can
More informationBasic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011
Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011 A. Miller 1. Introduction. The definitions of metric space and topological space were developed in the early 1900 s, largely
More information1. Prove that the empty set is a subset of every set.
1. Prove that the empty set is a subset of every set. Basic Topology Written by MenGen Tsai email: b89902089@ntu.edu.tw Proof: For any element x of the empty set, x is also an element of every set since
More informationChap2: The Real Number System (See Royden pp40)
Chap2: The Real Number System (See Royden pp40) 1 Open and Closed Sets of Real Numbers The simplest sets of real numbers are the intervals. We define the open interval (a, b) to be the set (a, b) = {x
More informationMATH 433 Applied Algebra Lecture 13: Examples of groups.
MATH 433 Applied Algebra Lecture 13: Examples of groups. Abstract groups Definition. A group is a set G, together with a binary operation, that satisfies the following axioms: (G1: closure) for all elements
More informationElements of probability theory
2 Elements of probability theory Probability theory provides mathematical models for random phenomena, that is, phenomena which under repeated observations yield di erent outcomes that cannot be predicted
More informationProblem Set. Problem Set #2. Math 5322, Fall December 3, 2001 ANSWERS
Problem Set Problem Set #2 Math 5322, Fall 2001 December 3, 2001 ANSWERS i Problem 1. [Problem 18, page 32] Let A P(X) be an algebra, A σ the collection of countable unions of sets in A, and A σδ the collection
More information5.1 Commutative rings; Integral Domains
5.1 J.A.Beachy 1 5.1 Commutative rings; Integral Domains from A Study Guide for Beginner s by J.A.Beachy, a supplement to Abstract Algebra by Beachy / Blair 23. Let R be a commutative ring. Prove the following
More informationMathematics Notes for Class 12 chapter 10. Vector Algebra
1 P a g e Mathematics Notes for Class 12 chapter 10. Vector Algebra A vector has direction and magnitude both but scalar has only magnitude. Magnitude of a vector a is denoted by a or a. It is nonnegative
More informationCH3 Boolean Algebra (cont d)
CH3 Boolean Algebra (cont d) Lecturer: 吳 安 宇 Date:2005/10/7 ACCESS IC LAB v Today, you ll know: Introduction 1. Guidelines for multiplying out/factoring expressions 2. ExclusiveOR and Equivalence operations
More information4. FIRST STEPS IN THE THEORY 4.1. A
4. FIRST STEPS IN THE THEORY 4.1. A Catalogue of All Groups: The Impossible Dream The fundamental problem of group theory is to systematically explore the landscape and to chart what lies out there. We
More informationLogic and Incidence Geometry
Logic and Incidence Geometry February 27, 2013 1 Informal Logic Logic Rule 0. No unstated assumption may be used in a proof. 2 Theorems and Proofs If [hypothesis] then [conclusion]. LOGIC RULE 1. The following
More informationFactoring, Solving. Equations, and Problem Solving REVISED PAGES
05W4801AM1.qxd 8/19/08 8:45 PM Page 241 Factoring, Solving Equations, and Problem Solving 5 5.1 Factoring by Using the Distributive Property 5.2 Factoring the Difference of Two Squares 5.3 Factoring
More informationProjective Geometry  Part 2
Projective Geometry  Part 2 Alexander Remorov alexanderrem@gmail.com Review Four collinear points A, B, C, D form a harmonic bundle (A, C; B, D) when CA : DA CB DB = 1. A pencil P (A, B, C, D) is the
More informationFuzzy Set Theory : Soft Computing Course Lecture 29 34, notes, slides RC Chakraborty, Aug.
Fuzzy Set Theory : Soft Computing Course Lecture 29 34, notes, slides www.myreaders.info/, RC Chakraborty, email rcchak@gmail.com, Aug. 10, 2010 http://www.myreaders.info/html/soft_computing.html www.myreaders.info
More informationMoufang planes with the Newton property
Moufang planes with the Newton property Zoltán Szilasi Abstract We prove that a Moufang plane with the Fano property satisfies the Newton property if and only if it is a Pappian projective plane. 1 Preliminaries
More informationSTRAIGHT LINES. , y 1. tan. and m 2. 1 mm. If we take the acute angle between two lines, then tan θ = = 1. x h x x. x 1. ) (x 2
STRAIGHT LINES Chapter 10 10.1 Overview 10.1.1 Slope of a line If θ is the angle made by a line with positive direction of xaxis in anticlockwise direction, then the value of tan θ is called the slope
More informationElementary Number Theory We begin with a bit of elementary number theory, which is concerned
CONSTRUCTION OF THE FINITE FIELDS Z p S. R. DOTY Elementary Number Theory We begin with a bit of elementary number theory, which is concerned solely with questions about the set of integers Z = {0, ±1,
More informationMA651 Topology. Lecture 6. Separation Axioms.
MA651 Topology. Lecture 6. Separation Axioms. This text is based on the following books: Fundamental concepts of topology by Peter O Neil Elements of Mathematics: General Topology by Nicolas Bourbaki Counterexamples
More informationFactoring Quadratic Expressions
Factoring the trinomial ax 2 + bx + c when a = 1 A trinomial in the form x 2 + bx + c can be factored to equal (x + m)(x + n) when the product of m x n equals c and the sum of m + n equals b. (Note: the
More informationA Natural Language Database Interface using Fuzzy Semantics. http://richard.bergmair.eu/
A Natural Language Database Interface using Fuzzy Semantics...wild speculation about the nature of truth, and other equally unscientific endeavours. http://richard.bergmair.eu/ Acknowledgments thanks for
More informationA set is an unordered collection of objects.
Section 2.1 Sets A set is an unordered collection of objects. the students in this class the chairs in this room The objects in a set are called the elements, or members of the set. A set is said to contain
More informationTheorem 5. The composition of any two symmetries in a point is a translation. More precisely, S B S A = T 2
Isometries. Congruence mappings as isometries. The notion of isometry is a general notion commonly accepted in mathematics. It means a mapping which preserves distances. The word metric is a synonym to
More informationFactoring Methods. Example 1: 2x + 2 2 * x + 2 * 1 2(x + 1)
Factoring Methods When you are trying to factor a polynomial, there are three general steps you want to follow: 1. See if there is a Greatest Common Factor 2. See if you can Factor by Grouping 3. See if
More information7  Linear Transformations
7  Linear Transformations Mathematics has as its objects of study sets with various structures. These sets include sets of numbers (such as the integers, rationals, reals, and complexes) whose structure
More informationTHE PRODUCT SPAN OF A FINITE SUBSET OF A COMPLETELY BOUNDED ARTEX SPACE OVER A BIMONOID
THE PRODUCT SPAN OF A FINITE SUBSET OF A COMPLETELY BOUNDED ARTEX SPACE OVER A BIMONOID ABSTRACT The product of subsets of an Artex space over a bimonoid is defined. Product Combination of elements of
More informationMAT2400 Analysis I. A brief introduction to proofs, sets, and functions
MAT2400 Analysis I A brief introduction to proofs, sets, and functions In Analysis I there is a lot of manipulations with sets and functions. It is probably also the first course where you have to take
More informationCourse 421: Algebraic Topology Section 1: Topological Spaces
Course 421: Algebraic Topology Section 1: Topological Spaces David R. Wilkins Copyright c David R. Wilkins 1988 2008 Contents 1 Topological Spaces 1 1.1 Continuity and Topological Spaces...............
More informationMetric Spaces. Chapter 1
Chapter 1 Metric Spaces Many of the arguments you have seen in several variable calculus are almost identical to the corresponding arguments in one variable calculus, especially arguments concerning convergence
More informationModule1. x 1000. y 800.
Module1 1 Welcome to the first module of the course. It is indeed an exciting event to share with you the subject that has lot to offer both from theoretical side and practical aspects. To begin with,
More informationMA 408 Computer Lab Two The Poincaré Disk Model of Hyperbolic Geometry. Figure 1: Lines in the Poincaré Disk Model
MA 408 Computer Lab Two The Poincaré Disk Model of Hyperbolic Geometry Put your name here: Score: Instructions: For this lab you will be using the applet, NonEuclid, created by Castellanos, Austin, Darnell,
More informationMATH INTRODUCTION TO PROBABILITY FALL 2011
MATH 755001 INTRODUCTION TO PROBABILITY FALL 2011 Lecture 4. Measurable functions. Random variables. I have told you that we can work with σalgebras generated by classes of sets in some indirect ways.
More informationDeterminants, Areas and Volumes
Determinants, Areas and Volumes Theodore Voronov Part 2 Areas and Volumes The area of a twodimensional object such as a region of the plane and the volume of a threedimensional object such as a solid
More informationUnit 3 Boolean Algebra (Continued)
Unit 3 Boolean Algebra (Continued) 1. ExclusiveOR Operation 2. Consensus Theorem Department of Communication Engineering, NCTU 1 3.1 Multiplying Out and Factoring Expressions Department of Communication
More information