Presentation is loading. Please wait.

Presentation is loading. Please wait.

CLASSICAL RELATIONS AND FUZZY RELATIONS

Published byAndra Butler Modified over 8 years ago

Similar presentations

Presentation on theme: "CLASSICAL RELATIONS AND FUZZY RELATIONS"— Presentation transcript:

1 CLASSICAL RELATIONS AND FUZZY RELATIONS
CHAPTER 8 CLASSICAL RELATIONS AND FUZZY RELATIONS “Principles of Soft Computing, 2nd Edition” by S.N. Sivanandam & SN Deepa Copyright  2011 Wiley India Pvt. Ltd. All rights reserved.

2 RELATIONS Relations represent mappings between sets and connectives in logic. A classical binary relation represents the presence or absence of a connection or interaction or association between the elements of two sets. Fuzzy binary relations are a generalization of crisp binary relations, and they allow various degrees of relationship (association) between elements. “Principles of Soft Computing, 2nd Edition” by S.N. Sivanandam & SN Deepa Copyright  2011 Wiley India Pvt. Ltd. All rights reserved.

3 CRISP CARTESIAN PRODUCT
“Principles of Soft Computing, 2nd Edition” by S.N. Sivanandam & SN Deepa Copyright  2011 Wiley India Pvt. Ltd. All rights reserved.

4 CRISP RELATIONS “Principles of Soft Computing, 2nd Edition”
by S.N. Sivanandam & SN Deepa Copyright  2011 Wiley India Pvt. Ltd. All rights reserved.

5 “Principles of Soft Computing, 2nd Edition”
by S.N. Sivanandam & SN Deepa Copyright  2011 Wiley India Pvt. Ltd. All rights reserved.

6 CRISP BINARY RELATIONS
Examples of binary relations “Principles of Soft Computing, 2nd Edition” by S.N. Sivanandam & SN Deepa Copyright  2011 Wiley India Pvt. Ltd. All rights reserved.

7 OPERATIONS ON CRISP RELATIONS
“Principles of Soft Computing, 2nd Edition” by S.N. Sivanandam & SN Deepa Copyright  2011 Wiley India Pvt. Ltd. All rights reserved.

8 PROPERTIES OF CRISP RELATIONS
The properties of crisp sets (given below) hold good for crisp relations as well. Commutativity, Associativity, Distributivity, Involution, Idempotency, DeMorgan’s Law, Excluded Middle Laws. “Principles of Soft Computing, 2nd Edition” by S.N. Sivanandam & SN Deepa Copyright  2011 Wiley India Pvt. Ltd. All rights reserved.

9 COMPOSITION ON CRISP RELATIONS
“Principles of Soft Computing, 2nd Edition” by S.N. Sivanandam & SN Deepa Copyright  2011 Wiley India Pvt. Ltd. All rights reserved.

10 FUZZY CARTESIAN PRODUCT
Let R be a fuzzy subset of M and S be a fuzzy subset of N. Then the Cartesian product R  S is a fuzzy subset of N  M such that Example: Let R be a fuzzy subset of {a, b, c} such that R = a/1 + b/0.8 + c/0.2 and S be a fuzzy subset of {1, 2, 3} such that S = 1/1 + 3/ /0.5. Then R x S is given by “Principles of Soft Computing, 2nd Edition” by S.N. Sivanandam & SN Deepa Copyright  2011 Wiley India Pvt. Ltd. All rights reserved.

11 FUZZY RELATION “Principles of Soft Computing, 2nd Edition”
by S.N. Sivanandam & SN Deepa Copyright  2011 Wiley India Pvt. Ltd. All rights reserved.

12 “Principles of Soft Computing, 2nd Edition”
by S.N. Sivanandam & SN Deepa Copyright  2011 Wiley India Pvt. Ltd. All rights reserved.

13 OPERATIONS ON FUZZY RELATION
The basic operation on fuzzy sets also apply on fuzzy relations. “Principles of Soft Computing, 2nd Edition” by S.N. Sivanandam & SN Deepa Copyright  2011 Wiley India Pvt. Ltd. All rights reserved.

14 “Principles of Soft Computing, 2nd Edition”
by S.N. Sivanandam & SN Deepa Copyright  2011 Wiley India Pvt. Ltd. All rights reserved.

15 PROPERTIES OF FUZZY RELATIONS
The properties of fuzzy sets (given below) hold good for fuzzy relations as well. Commutativity, Associativity, Distributivity, Involution, Idempotency, DeMorgan’s Law, Excluded Middle Laws. “Principles of Soft Computing, 2nd Edition” by S.N. Sivanandam & SN Deepa Copyright  2011 Wiley India Pvt. Ltd. All rights reserved.

16 COMPOSITION OF FUZZY RELATIONS
“Principles of Soft Computing, 2nd Edition” by S.N. Sivanandam & SN Deepa Copyright  2011 Wiley India Pvt. Ltd. All rights reserved.

17 “Principles of Soft Computing, 2nd Edition”
by S.N. Sivanandam & SN Deepa Copyright  2011 Wiley India Pvt. Ltd. All rights reserved.

18 “Principles of Soft Computing, 2nd Edition”
by S.N. Sivanandam & SN Deepa Copyright  2011 Wiley India Pvt. Ltd. All rights reserved.

19 “Principles of Soft Computing, 2nd Edition”
by S.N. Sivanandam & SN Deepa Copyright  2011 Wiley India Pvt. Ltd. All rights reserved.

20 CLASSICAL EQUIVALENCE RELATION
“Principles of Soft Computing, 2nd Edition” by S.N. Sivanandam & SN Deepa Copyright  2011 Wiley India Pvt. Ltd. All rights reserved.

21 CLASSICAL TOLERANCE RELATION
“Principles of Soft Computing, 2nd Edition” by S.N. Sivanandam & SN Deepa Copyright  2011 Wiley India Pvt. Ltd. All rights reserved.

22 FUZZY EQUIVALENCE RELATION
“Principles of Soft Computing, 2nd Edition” by S.N. Sivanandam & SN Deepa Copyright  2011 Wiley India Pvt. Ltd. All rights reserved.

23 FUZZY TOLERANCE RELATION
A binary fuzzy relation that possesses the properties of reflexivity and symmetry is called fuzzy tolerance relation or resemblance relation. The equivalence relations are a special case of the tolerance relation. The fuzzy tolerance relation can be reformed into fuzzy equivalence relation in the same way as a crisp tolerance relation is reformed into crisp equivalence relation, i.e., where ‘n’ is the cardinality of the set that defines R1. “Principles of Soft Computing, 2nd Edition” by S.N. Sivanandam & SN Deepa Copyright  2011 Wiley India Pvt. Ltd. All rights reserved.

24 SUMMARY The chapter provides the definition, properties, and operations of the following: Classical sets, Fuzzy sets, Classical relations, Fuzzy relations. The chapter also discusses crisp and fuzzy equivalence and tolerance relations. “Principles of Soft Computing, 2nd Edition” by S.N. Sivanandam & SN Deepa Copyright  2011 Wiley India Pvt. Ltd. All rights reserved.

Download ppt "CLASSICAL RELATIONS AND FUZZY RELATIONS"

Similar presentations

Union Definition: The union of sets A and B, denoted by A B, contains those elements that are in A or B or both: Example: { 1, 2, 3} {3, 4, 5} = { 1,

Union Definition: The union of sets A and B, denoted by A B, contains those elements that are in A or B or both: Example: { 1, 2, 3} {3, 4, 5} = { 1,
Fuzzy Logic (1) Intelligent System Course. Apples, oranges or in between? A O Group of appleGroup of orange A A AA A A O O O O O O.

Fuzzy Logic (1) Intelligent System Course. Apples, oranges or in between? A O Group of appleGroup of orange A A AA A A O O O O O O.
Presented by: Ms. Maria Estrellita D. Hechanova, ECE

Presented by: Ms. Maria Estrellita D. Hechanova, ECE
Computational Intelligence Winter Term 2014/15 Prof. Dr. Günter Rudolph Lehrstuhl für Algorithm Engineering (LS 11) Fakultät für Informatik TU Dortmund.

Computational Intelligence Winter Term 2014/15 Prof. Dr. Günter Rudolph Lehrstuhl für Algorithm Engineering (LS 11) Fakultät für Informatik TU Dortmund.
Fuzziness vs. Probability JIN Yan Nov. 17, 2004. The outline of Chapter 7 Part I Fuzziness vs. probability Part II Fuzzy sets & relevant theories.

Fuzziness vs. Probability JIN Yan Nov. 17, The outline of Chapter 7 Part I Fuzziness vs. probability Part II Fuzzy sets & relevant theories.
Fuzzy Sets and Applications Introduction Introduction Fuzzy Sets and Operations Fuzzy Sets and Operations.

Fuzzy Sets and Applications Introduction Introduction Fuzzy Sets and Operations Fuzzy Sets and Operations.
CS621: Introduction to Artificial Intelligence Pushpak Bhattacharyya CSE Dept., IIT Bombay Lecture–3: Some proofs in Fuzzy Sets and Fuzzy Logic 27 th July.

CS621: Introduction to Artificial Intelligence Pushpak Bhattacharyya CSE Dept., IIT Bombay Lecture–3: Some proofs in Fuzzy Sets and Fuzzy Logic 27 th July.
The Engineering Design of Systems: Models and Methods

The Engineering Design of Systems: Models and Methods
1 Section 1.7 Set Operations. 2 Union The union of 2 sets A and B is the set containing elements found either in A, or in B, or in both The denotation.

1 Section 1.7 Set Operations. 2 Union The union of 2 sets A and B is the set containing elements found either in A, or in B, or in both The denotation.
Relations. Binary Relations a relation between elements of two sets is a subset of their Cartesian product (of ordered pairs). Note the di ff erence between.

Relations. Binary Relations a relation between elements of two sets is a subset of their Cartesian product (of ordered pairs). Note the di ff erence between.
1 Section 10.1 Boolean Functions. 2 Computers & Boolean Algebra Circuits in computers have inputs whose values are either 0 or 1 Mathematician George.

1 Section 10.1 Boolean Functions. 2 Computers & Boolean Algebra Circuits in computers have inputs whose values are either 0 or 1 Mathematician George.
CLASSICAL RELATIONS AND FUZZY RELATIONS

CLASSICAL RELATIONS AND FUZZY RELATIONS
Theory and Applications

Theory and Applications
Fuzzy Logic B. Logic Operations. Conventional or crisp sets are binary. An element either belongs to the set or doesn't. Fuzzy sets, on the other hand,

Fuzzy Logic B. Logic Operations. Conventional or crisp sets are binary. An element either belongs to the set or doesn't. Fuzzy sets, on the other hand,

Algebra Algebra – defined by the tuple:  A, o 1, …, o k ; R 1, …, R m ; c 1, …, c k  Where: A is a non-empty set o i is the function, o i : A p i  A.

Algebra Algebra – defined by the tuple:  A, o 1, …, o k ; R 1, …, R m ; c 1, …, c k  Where: A is a non-empty set o i is the function, o i : A p i  A.
Classical Relations and Fuzzy Relations

Classical Relations and Fuzzy Relations
Classical Sets and Fuzzy Sets

Classical Sets and Fuzzy Sets
International Certification Services Pvt Ltd. www.icsasian.com.

International Certification Services Pvt Ltd.
Boolean Algebra and Computer Logic Mathematical Structures for Computer Science Chapter 7.1 – 7.2 Copyright © 2006 W.H. Freeman & Co.MSCS Slides Boolean.

Boolean Algebra and Computer Logic Mathematical Structures for Computer Science Chapter 7.1 – 7.2 Copyright © 2006 W.H. Freeman & Co.MSCS Slides Boolean.

Similar presentations

Ads by Google