Chapter 3 FUZZY RELATION AND COMPOSITION

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Chapter 3 FUZZY RELATION AND COMPOSITION

• G.Anuradha

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Outline

• Product set
• Crisp / fuzzy relations
• Composition / decomposition
• Projection / cylindrical extension
• Extension of fuzzy set / fuzzy relation

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Product set
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Product set
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Product set

• Aa1,a2 Bb1,b2 Cc1,c2
• AxBxC (a1,b1,c1),(a1,b1,c2),(a1,b2,c1),(a1,b2,c
  2),(a2,b1,c1),(a2,b1,c2),(a2,b2,c1), (a2,b2,c2)

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Crisp relation

• A relation among crisp sets
  is a subset of the Cartesian product. It is
  denoted by .
• Using the membership function defines the crisp
  relation R

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Fuzzy relation

• A fuzzy relation is a fuzzy set defined on the
  Cartesian product of crisp sets A1, A2, ..., An
  where tuples (x1, x2, ..., xn) may have varying
  degrees of membership within the relation.
• The membership grade indicates the strength of
  the relation present between the elements of the
  tuple.

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Representation methods

• Bipartigraph

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Representation methods

• Matrix

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Representation methods

• Digraph

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Domain and range of fuzzy relation

• Domain
• Range

domain
range
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Domain and range of fuzzy relation

• Fuzzy matrix

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Operations on fuzzy matrices

• Sum
• Example

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Operations on fuzzy matrices

• Max product C ABAB
• Example

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Max product

• Example

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Max product

• Example

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Max product

• Example

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Operations on fuzzy matrices

• Scalar product
• Example

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Operations on fuzzy relations

• Union relation
• For n relations

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Union relation

• Example

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Operations on fuzzy relations

• Intersection relation
• For n relations

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Intersection relation

• Example

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Operations on fuzzy relations

• Complement relation
• Example

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Composition of fuzzy relations

• Max-min composition
• Example

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Composition of fuzzy relations
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Composition of fuzzy relations

• Example

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Composition of fuzzy relations

• Example

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Composition of fuzzy relations
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a-cut of fuzzy relation

• Example

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a-cut of fuzzy relation
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Decomposition of relation
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Decomposition of relation
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Decomposition of relation
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Projection / cylindrical extension
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Projection / cylindrical extension
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Projection in n dimension
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Projection
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Projection
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Projection
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Projection
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Projection / cylindrical extension
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Cylindrical extension
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Cylindrical extension
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Cylindrical extension

• x1 0 x,x1 1 y
• x2 0 a, x2 1 b
• x3 0 a, x3 1 ß

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Cylindrical extension

• Join(R123,R123) C(R123)nC(R123)
• Min(R123,R123)
• R123

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Functions with Fuzzy Arguments

• A crisp function maps its crisp input argument to
  its image.
• Fuzzy arguments have membership degrees.
• When computing a fuzzy mapping it is necessary
  to compute the image and its membership value.

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Crisp Mappings
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Other operations on fuzzy sets

• Cartesian product
• Mth power
• Algebraic sum
• Bounded sum
• Bounded difference
• Algebraic product

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Cartesian product

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Thanks for your attention!