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Soft ComputingFuzzy Relations

Prof. Debasis SamantaDepartment of Computer Science & Engineering

IIT Kharagpur

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Fuzzy Relations

• Crisp relatons

• Operatons on crisp relatons

• Examples on crisp relatons

• Fuzzy relatons

• Operatons on fuzzy relatons

• Examples on fuzzy relatons

Debasis SamantaCSE

IIT Kharagpur

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Crisp relations

• Order pairs:

Suppose, A and B are two (crisp) sets. Then Cartesian product denoted as is a collecton of order pairs, such that

Note :

(1) (2)

(3) provides a mapping from to .

A partcular mapping so mentoned is called a relaton.

Debasis SamantaCSE

IIT Kharagpur

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Crisp relations

Example:

Consider the two crisp sets and as given below. } .

Then, ,

Let us defne a relaton as }

Then, in this case.

Debasis SamantaCSE

IIT Kharagpur

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Crisp relations

We can represent the relaton in a matrix form as follows.

Debasis SamantaCSE

IIT Kharagpur

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Operations on crisp relationsSuppose, and are the two relatons defned over two crisp sets and

• Union:

• Intersecton:

• Complement:

Debasis SamantaCSE

IIT Kharagpur

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Example: Operations on crisp relationsSuppose, and are the two relatons defned over two crisp sets and

Find the following

Debasis SamantaCSE

IIT Kharagpur

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Composition of two crisp relationsGiven is a relaton on and is another relaton on . Then, is called a compositon of relaton on and which is defned as follows.

Max-Min Compositon

Given the two relaton matrices and , the max-min compositon is defned as ;

Debasis SamantaCSE

IIT Kharagpur

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Composition: CompositionExample : Given

Here, and is on .

Thus, we have

Using max-min compositon

Debasis SamantaCSE

IIT Kharagpur

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Fuzzy relations• Fuzzy relaton is a fuzzy set defned on the Cartesian product of crisp set

• Here, n-tuples may have varying degree of memberships within the relatonship.

• The membership values indicate the strength of the relaton between the tuples.

Debasis SamantaCSE

IIT Kharagpur

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Fuzzy relations

Example:

The fuzzy relaton is defned as

Debasis SamantaCSE

IIT Kharagpur

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Fuzzy Cartesian product

Suppose

• is a fuzzy set on the universe of discourse with

• is a fuzzy set on the universe of discourse with

Then ; where has its membership functon given by

Debasis SamantaCSE

IIT Kharagpur

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Fuzzy Cartesian product

Example :

and

Debasis SamantaCSE

IIT Kharagpur

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Operations on Fuzzy relations

Let and be two fuzzy relatons on .

• Union:

• Intersecton:

• Complement:

• Compositon:

Debasis SamantaCSE

IIT Kharagpur

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Operations on Fuzzy relations: Example

Example :

Debasis SamantaCSE

IIT Kharagpur

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Fuzzy relation : An exampleConsider the following two sets and which represent a set of paddy plants and a set of plant diseases. More precisely

a set of four varietes of paddy plantsof the four various diseases afectng the plants.

In additon to these, also consider another set be the common symptoms of the diseases. Let, be a relaton on , representng which plant is susceptble to which diseases, which is stated as

Debasis SamantaCSE

IIT Kharagpur

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Fuzzy relation : An exampleAlso, consider be the another relaton on , which is given by

Obtain the associaton of plants with the diferent symptoms of the disease using max-min compositon.Hint: Find , and verify that

Debasis SamantaCSE

IIT Kharagpur

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Fuzzy relation : Another exampleLet, is relevant to and is relevant to be two fuzzy relatons defned on and , respectvely, where , and Assume that and can be expressed with the following relaton matrices :

Debasis SamantaCSE

IIT Kharagpur

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Fuzzy relation : Another exampleNow, we want to fnd , which can be interpreted as a derived fuzzy relaton is relevant to .Suppose, we are only interested in the degree of relevance between and . Then, using max-min compositon,

Debasis SamantaCSE

IIT Kharagpur

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2D Membership functions : Binary fuzzy relations(Binary) fuzzy relatons are fuzzy sets which map each element in to a

membership grade between 0 and 1 (both inclusive). Note that a membership functon of a binary fuzzy relaton can be depicted with a 3D plot.

Important: Binary fuzzy relatons are fuzzy sets with two dimensional MFs and so on.

Debasis SamantaCSE

IIT Kharagpur

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2D membership function : An exampleLet(the positve real line) and is much greater than

The membership functon of is defned as

Suppose, and , then

Debasis SamantaCSE

IIT Kharagpur

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Example:How you can derive the following?If x is A or y is B then z is C;Given that• R1: If x is A then z is • R2: If y is B then z is

Hint: You have given two relatons R1 and R2. Then, the required can be derived using the union operaton of R1

and R2

Debasis SamantaCSE

IIT Kharagpur

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Thank You!!

Debasis SamantaCSE

IIT Kharagpur