1 of 19

## What's hot

Fuzzy logic Notes AI CSE 8th Sem
Fuzzy logic Notes AI CSE 8th SemDigiGurukul

Fuzzy logic and application in AI
Fuzzy logic and application in AIIldar Nurgaliev

Decision tree, softmax regression and ensemble methods in machine learning
Decision tree, softmax regression and ensemble methods in machine learningAbhishek Vijayvargia

Constraint satisfaction problems (csp)
Constraint satisfaction problems (csp) Archana432045

State Space Search in ai
State Space Search in aivikas dhakane

Extension principle
Extension principleSavo Delić

Optimization in Deep Learning
Optimization in Deep LearningYan Xu

Artificial Intelligence Searching Techniques
Artificial Intelligence Searching TechniquesDr. C.V. Suresh Babu

8. R Graphics with R
8. R Graphics with RFAO

AI 7 | Constraint Satisfaction Problem
AI 7 | Constraint Satisfaction ProblemMohammad Imam Hossain

Medians and order statistics
Medians and order statisticsRajendran

Fuzzy rules and fuzzy reasoning
Fuzzy rules and fuzzy reasoningVeni7

Data Science - Part XII - Ridge Regression, LASSO, and Elastic Nets
Data Science - Part XII - Ridge Regression, LASSO, and Elastic NetsDerek Kane

### What's hot(20)

Fuzzy logic Notes AI CSE 8th Sem
Fuzzy logic Notes AI CSE 8th Sem

Fuzzy logic and application in AI
Fuzzy logic and application in AI

Decision tree, softmax regression and ensemble methods in machine learning
Decision tree, softmax regression and ensemble methods in machine learning

L9 fuzzy implications
L9 fuzzy implications

Constraint satisfaction problems (csp)
Constraint satisfaction problems (csp)

State Space Search in ai
State Space Search in ai

Extension principle
Extension principle

Optimization in Deep Learning
Optimization in Deep Learning

lattice
lattice

Artificial Intelligence Searching Techniques
Artificial Intelligence Searching Techniques

8. R Graphics with R
8. R Graphics with R

AI 7 | Constraint Satisfaction Problem
AI 7 | Constraint Satisfaction Problem

First order logic
First order logic

Medians and order statistics
Medians and order statistics

First order logic
First order logic

L3 some other properties
L3 some other properties

Fuzzy rules and fuzzy reasoning
Fuzzy rules and fuzzy reasoning

Fuzzy logic ppt
Fuzzy logic ppt

Data Science - Part XII - Ridge Regression, LASSO, and Elastic Nets
Data Science - Part XII - Ridge Regression, LASSO, and Elastic Nets

Introduction to soft computing
Introduction to soft computing

## Similar to Rough Set and Fuzzy Set Theory Comparison

Fuzzy Set Theory
Fuzzy Set TheoryAMIT KUMAR

Unit-II -Soft Computing.pdf
Unit-II -Soft Computing.pdfRamya Nellutla

53158699-d7c5-4e6e-af19-1f642992cc58-161011142651.pptx

Fuzzy Logic.pptx
Fuzzy Logic.pptxImXaib

Fuzzy logic-introduction
Fuzzy logic-introductionWBUTTUTORIALS

E5-roughsets unit-V.pdf
E5-roughsets unit-V.pdfRamya Nellutla

Fuzzy random variables and Kolomogrov’s important results
Fuzzy random variables and Kolomogrov’s important resultsinventionjournals

Cs229 cvxopt
Cs229 cvxoptcerezaso

Optimization using soft computing
Optimization using soft computingPurnima Pandit

Introduction to Artificial Intelligence
Introduction to Artificial IntelligenceManoj Harsule

IVR - Chapter 1 - Introduction
IVR - Chapter 1 - IntroductionCharles Deledalle

Probability and Statistics
Probability and StatisticsMalik Sb

### Similar to Rough Set and Fuzzy Set Theory Comparison(20)

Fuzzy Set Theory
Fuzzy Set Theory

Unit-II -Soft Computing.pdf
Unit-II -Soft Computing.pdf

53158699-d7c5-4e6e-af19-1f642992cc58-161011142651.pptx
53158699-d7c5-4e6e-af19-1f642992cc58-161011142651.pptx

Fuzzy Logic.pptx
Fuzzy Logic.pptx

Fuzzy logic-introduction
Fuzzy logic-introduction

FUZZY COMPLEMENT
FUZZY COMPLEMENT

Fuzzy Logic_HKR
Fuzzy Logic_HKR

Fuzzylogic
Fuzzylogic

E5-roughsets unit-V.pdf
E5-roughsets unit-V.pdf

Fuzzy random variables and Kolomogrov’s important results
Fuzzy random variables and Kolomogrov’s important results

Cs229 cvxopt
Cs229 cvxopt

Optimization using soft computing
Optimization using soft computing

Program on Quasi-Monte Carlo and High-Dimensional Sampling Methods for Applie...
Program on Quasi-Monte Carlo and High-Dimensional Sampling Methods for Applie...

Introduction to Artificial Intelligence
Introduction to Artificial Intelligence

PredicateLogic (1).ppt
PredicateLogic (1).ppt

PredicateLogic.pptx
PredicateLogic.pptx

L7 fuzzy relations
L7 fuzzy relations

Per6 basis2_NUMBER SYSTEMS
Per6 basis2_NUMBER SYSTEMS

IVR - Chapter 1 - Introduction
IVR - Chapter 1 - Introduction

Probability and Statistics
Probability and Statistics

## More from AMIT KUMAR

MultiObjective(11) - Copy
MultiObjective(11) - CopyAMIT KUMAR

soft computing
soft computingAMIT KUMAR

Soft Computing-173101
Soft Computing-173101AMIT KUMAR

RESEARCH METHODOLOGY
RESEARCH METHODOLOGYAMIT KUMAR

### More from AMIT KUMAR(8)

MultiObjective(11) - Copy
MultiObjective(11) - Copy

final seminar
final seminar

EJSR(5)
EJSR(5)

coa dea(3)
coa dea(3)

1641
1641

soft computing
soft computing

Soft Computing-173101
Soft Computing-173101

RESEARCH METHODOLOGY
RESEARCH METHODOLOGY

### Rough Set and Fuzzy Set Theory Comparison

• 2. FuzzyFuzzy SetsSets • Dr. Lotfi Zadeh propose this approach. • In his approach an element can belong to a degree k (0 <= k <= 1). • In classical set theory an element must belong or not belong to a set. • Fuzzy membership function can be presented as: µX(x) є (0,1) where, X is a set and x is an element. 2
• 3. Fuzzy Sets (Continue)Fuzzy Sets (Continue) • Fuzzy membership function has the following properties. • µU -x(x) = 1 - µX(x) for any x є U • µxUy(x) = max(µX(x), µy(x)) for any x є U • µx ∩y(x) = min(µX(x), µy(x)) for any x є U 3
• 4. Rough SetsRough Sets • Rough set theory is another approach to handle vagueness. • Imprecision in this approach is expressed by a boundary region of a set, and not by a partial membership, like in fuzzy set theory. • Rough set concept can be defined by topological operation interior and closure called approximations. 4
• 5. Rough Sets (Continue)Rough Sets (Continue) • Suppose we are given a set of objects U called the universe and an indiscernibility relation R ⊆ U × U, representing our lack of knowledge about elements of U. • For simplicity we assume that R is an equivalence relation. • Let X be a subset of U. • We want to characterize the set X with respect to R. To do this we will need the basic concepts of rough set theory given in next slide. 5
• 6. Rough Sets (Continue)Rough Sets (Continue) • The lower approximation of a set X with respect to R is the set of all objects, which can be for certain(sure) classified as X with respect to R (are certainly X with respect to R). • The upper approximation of a set X with respect to R is the set of all objects which can be possibly(maybe) classified as X with respect to R (are possibly X in view of R). • The boundary region of a set X with respect to R is the set of all objects, which can be classified neither as X nor as not-X with respect to R. 6
• 7. Rough Sets (Continue)Rough Sets (Continue) • So that, • Set X is crisp (Exact with respect to R), if the boundary region of X is empty. • Set X is rough (Inexact with respect to R), if the boundary region of X is nonempty. 7
• 8. Rough ApproximationRough Approximation • Formal definitions of approximations and the boundary region are as follows: • R-lower approximation of X R*(x) = U {R(x): R(x) ⊆ X} • R- upper approximation of X R* (x) = U {R(x): R(x) ∩ X ≠ ɸ} • R-boundary approximation of X RNR (X) = R* (X) - R* (X) 8
• 9. Rough ApproximationRough Approximation • As we can see from the definition, approximations are expressed in terms of granules of knowledge. • The lower approximation of a set is union of all granules which are entirely included in the set. • The upper approximation − is union of all granules which have non-empty intersection with the set. • The boundary region of set is the difference between the upper and the lower approximation. 9
• 10. Rough Membership functionRough Membership function • Rough sets can be also defined as rough membership function. µx R : U  (0,1) Where µx R (x) = |X∩ R(x)| / |R(x)| And |X| denotes the cardinality of X. 10
• 11. Rough MembershipRough Membership function(Conti)function(Conti) • The rough membership function can be expresses conditional probability. • That x belongs to X given R. • And can be interpreted as a degree that x belongs to X in view of information about x expressed by R. • The meaning of rough membership function can be defined as shown in fig 1. 11
• 13. Rough MembershipRough Membership function(Conti)function(Conti) • The rough membership function can be used to define approximations and the boundary region of a set, as shown below: R* (x) = {xєU : µx R (x) = 1} R* (x) = {xєU : µx R (x) > 0} RNR (X) = {xєU : 0 < µx R (x) < 1 } 13
• 14. Rough MembershipRough Membership function(Conti)function(Conti) • The rough membership function has the following properties: • µx R (x) = 1 iff x є R* (x) • µx R (x) = 0 iff x є U - R* (x) • 0 < µx R (x) < 1 iff x є RNR (X) • µR U-x (x) = 1 - µx R (x) for any x є U • µxUy(x) => max(µR X(x), µR y(x)) for any x є U • µx ∩y(x) <= min (µR X(x), µR y(x)) for any x є U 14
• 15. IndiscernibilityIndiscernibility • A decision system (i.e. a decision table) express all the model. • This table may be unnecessarily large. • The same or indiscernible objects may be represented several times. 15
• 16. Indiscernibility (Conti)Indiscernibility (Conti) Element Age LEMS Walk X1 16-30 50 Yes X2 16-30 0 No X3 31-45 1-25 No X4 31-45 1-25 Yes X5 46-60 26-49 No X6 16-30 26-49 No X7 46-60 26-49 Yes LEMS = Lower Extremity(boundary) Motor Score Table 1 16
• 17. IndiscernibilityIndiscernibility • A binary relation R ⊆ X x X which is reflexive (i.e. an object is in relation with itself xRx), symmetric (if xRy then yRx) and transitive (if xRy and yRz then xRz) is called an equivalence relation. • The equivalence class of an element x є X consists of all objects y є X such that xRy. • Let A = (U, A) be an information system then with any B ⊆ A there is associated an equivalence relation INDA (B) : INDA (B) = {(x,x’) є U2 | a є B a(x) = a(x’)} 17
• 18. IndiscernibilityIndiscernibility • INDA (B) is called the B-indiscernibility relation. If (x,x') є INDA (B) then x and x' are indiscernible from each other by attributes from B. • The equivalence classes of the B-indiscernibility relation are denoted [x]B. Example: • Let us illustrate how a decision table such as Table 1 defines an indiscernibility relation. • The non-empty subsets of the conditional attributes are {Age}, {LEMS} and {Age, LEMS}. 18
• 19. IndiscernibilityIndiscernibility • If we consider, for instance, {LEMS}, objects X3 and X4 belong to the same equivalence class and are indiscernible. • The relation IND defines three partitions of the universe. IND ({Age}) = {{x1,x2,x6},{x3,x4},{x5,x7}} IND ({LEMS}) = {{x1},{x2},{x6},{x3,x4},{x5,x6,x7}} IND ({Age,LEMS}) = {{x1},{x2},x6},{x3,x4},{x5,x7},{x6}} 19
Current LanguageEnglish
Español
Portugues
Français
Deutsche