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BANGALORE UNIVERSITY
UNIVERSITY VISVESVARAYA COLLEGE OF ENGINEERING
K R Circle, Bangalore 560001
Department of Computer Science and Engineering
COMPUTER NETWORKING
Seminar On
FUZZY RELATIONS
Submitted By:
VAISHALI BAGEWADIKAR
20GACS4017
Dept of CSE, CN Branch.
2021-2022
Under the Guidance of:
Dr. PRATIBHAVANI .P.M
Associate Professor
Dept of CSE,UVCE
Bangalore
Agenda
• Crisp sets
• Fuzzy sets
• Cartesian product
• Crisp relation
• Fuzzy Relations
Basics
• Crisp set
• A set defined using a characteristic function that assigns a value of
either 0 or 1 to each element of the universe, discriminating
between members and non-members of the crisp set under
consideration.
• Example : light is ON or OFF
• In the context of fuzzy sets theory, crisp sets are referred as
“classical” or “ordinary” sets
• Fuzzy sets are generalization of crisp sets where the degree of
inclusiveness of an element may be anything from 0 to 1. Not just 0
or 1.
• Example: weather is very cold.
• Definition Fuzzy set
A fuzzy set F on a given universe of discourse U is defined as a
collection of ordered pairs (x, μF (x)) where x ∊ U, and for all x ∊ U,
0.0 ≤ μF (x) ≤ 1.0.
F = {(x, μF (x))} | x ∊ U, 0.0 ≤ μF (x) ≤ 1.0}
Definition of Cartesian product
Let A and B be two sets. Then the Cartesian product of A and B,
denoted by A × B, is the set of all ordered pairs (a, b) such that
a ∊ A, and b ∊ B.
A × B = {(a, b) | a ∊ A, and b ∊ B} Since (a, b) ≠ (b, a) we have
in general A × B ≠ B × A. Hence the operation of Cartesian
product is not commutative
Cartesian product
Definition -Membership Function
• Given an element x and a set S, the membership of x with respect
to S, denoted as μ S (x), is defined as :
• µS (x) = 1, if x ∈ S
• µS (x) = 0, if x ∉ S
Definition -Crisp relation
• Given two crisp sets A and B, a crisp relation R between A and B is a subset of
A × B. and R ⊆ A × B
• Consider the sets A = {1, 2, 3}, B = {1, 2, 3, 4}
relation R = {(a, b) | b = a + 1, a ∊ A, and b ∊B}.
Then R = {(1, 2), (2, 3), (3, 4)}.
• Here R ⊂ A × B.
• A crisp relation between sets A and B is expressed with the help of a relation
matrix T.
Example
The rows and the columns of the relation matrix T correspond to
the members of A and B respectively.
A = {1, 2, 3}, B = {1, 2, 3, 4}
relation R = {(a, b) | b = a + 1, a ∊ A, and b ∊ B}
R = {(1, 2), (2, 3), (3, 4)}. Relation matrix for R is given below
EXAMPLE 2
EXAMPLE 3
Defnition:Fuzzy Cartesian product
Cardinality of Fuzzy Relations
Operations On Fuzzy Relation
μ
Example
Properties of Fuzzy Relations
• The properties of fuzzy sets hold good for fuzzy relations as well.
 Commutativity
 Associativity
 Distributivity
 Involution
 Idempotency
 DeMorgan’s Law
 Excluded Middle Laws.
Fuzzy Composition
Example(Max-Min )
= max [ min (0.6, 1),min (0.3,0.8)]
= max [0.6, 0.3]
= 0.6
=max [min (0.6,0.5),min (0.3,0.4)]
=max [min (0.6,0.5),min (0.3,0.4)]
= max (0.5, 0.3)
= 0.5
max [min (0.6,0.3),min (0.3,0.7)]
MT(x1,z3)=max [min (0.6,0.3),min (0.3,0.7)]
= max [0.3, 0.3] = 0.3
MT(x2,z1)=max [min (0.2,1),min (0.9,0.8)]
MT(x2,z1)=max [min (0.2,1),min (0.9,0.8)]
= max [ 0.2, 0.8] = 0.8
=max [min (0.2,0.5),min (0.3,0.4)]
MT(x2,z2)=max [min (0.2,0.5),min (0.3,0.4)]
= max [0.2, 0.4] = 0.4
max [min (0.2,0.3),min (0.9,0.7)]
MT(x2,z3)=max [min (0.2,0.3),min (0.9,0.7)]
= max (0.2, 0.7) = 0.7
T = RoS = [0.6 0.5 0.3
0.8 0.4 0.7]
Example(Max-Prod)
Example(Max-Prod)
T = R . S
MT(x1,z1) = max [MR(x1,y1).MS(y1,z1)]
= max [MR(x1,y1).MS(y1,z1)] MR(x1,y2).MS(y2,z1)]
MR(x1,y2).MS(y2,z1)]= max (0.6, 0.24) = 0.6
MT(x1,z2)=max [MR(x1,y1).MS(y1,z2)]
=max [MR(x1,y1).MS(y1,z2)]
[MR(x1,y2).MS(y2,z2)][MR(x1,y2).MS(y2,z2)]
= max [0.30, 0.12] = 0.30
Example(Max-Prod)
MT(x1,z3)=max [0.18,0.21]=0.21
MT(x2,z1)=max [0.2,0.72]=0.72
MT(x2,z2)=max [0.10,0.36]=0.36
MT(x2,z3)=max [0.06,0.63]=0.63
z1 z2 z3
T= R.S = x1 [ 0.6 0.3 0.21 ]
x2 [ 0.72 0.36 0.63]
FuzzyRelations.pptx

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FuzzyRelations.pptx

  • 1. BANGALORE UNIVERSITY UNIVERSITY VISVESVARAYA COLLEGE OF ENGINEERING K R Circle, Bangalore 560001 Department of Computer Science and Engineering COMPUTER NETWORKING Seminar On FUZZY RELATIONS Submitted By: VAISHALI BAGEWADIKAR 20GACS4017 Dept of CSE, CN Branch. 2021-2022 Under the Guidance of: Dr. PRATIBHAVANI .P.M Associate Professor Dept of CSE,UVCE Bangalore
  • 2. Agenda • Crisp sets • Fuzzy sets • Cartesian product • Crisp relation • Fuzzy Relations
  • 3. Basics • Crisp set • A set defined using a characteristic function that assigns a value of either 0 or 1 to each element of the universe, discriminating between members and non-members of the crisp set under consideration. • Example : light is ON or OFF • In the context of fuzzy sets theory, crisp sets are referred as “classical” or “ordinary” sets • Fuzzy sets are generalization of crisp sets where the degree of inclusiveness of an element may be anything from 0 to 1. Not just 0 or 1. • Example: weather is very cold.
  • 4. • Definition Fuzzy set A fuzzy set F on a given universe of discourse U is defined as a collection of ordered pairs (x, μF (x)) where x ∊ U, and for all x ∊ U, 0.0 ≤ μF (x) ≤ 1.0. F = {(x, μF (x))} | x ∊ U, 0.0 ≤ μF (x) ≤ 1.0}
  • 5. Definition of Cartesian product Let A and B be two sets. Then the Cartesian product of A and B, denoted by A × B, is the set of all ordered pairs (a, b) such that a ∊ A, and b ∊ B. A × B = {(a, b) | a ∊ A, and b ∊ B} Since (a, b) ≠ (b, a) we have in general A × B ≠ B × A. Hence the operation of Cartesian product is not commutative
  • 7. Definition -Membership Function • Given an element x and a set S, the membership of x with respect to S, denoted as μ S (x), is defined as : • µS (x) = 1, if x ∈ S • µS (x) = 0, if x ∉ S
  • 8. Definition -Crisp relation • Given two crisp sets A and B, a crisp relation R between A and B is a subset of A × B. and R ⊆ A × B • Consider the sets A = {1, 2, 3}, B = {1, 2, 3, 4} relation R = {(a, b) | b = a + 1, a ∊ A, and b ∊B}. Then R = {(1, 2), (2, 3), (3, 4)}. • Here R ⊂ A × B. • A crisp relation between sets A and B is expressed with the help of a relation matrix T.
  • 9. Example The rows and the columns of the relation matrix T correspond to the members of A and B respectively. A = {1, 2, 3}, B = {1, 2, 3, 4} relation R = {(a, b) | b = a + 1, a ∊ A, and b ∊ B} R = {(1, 2), (2, 3), (3, 4)}. Relation matrix for R is given below
  • 10.
  • 11.
  • 12.
  • 13.
  • 14.
  • 15.
  • 16.
  • 18.
  • 21. Cardinality of Fuzzy Relations
  • 22. Operations On Fuzzy Relation μ
  • 24. Properties of Fuzzy Relations • The properties of fuzzy sets hold good for fuzzy relations as well.  Commutativity  Associativity  Distributivity  Involution  Idempotency  DeMorgan’s Law  Excluded Middle Laws.
  • 27. = max [ min (0.6, 1),min (0.3,0.8)] = max [0.6, 0.3] = 0.6 =max [min (0.6,0.5),min (0.3,0.4)] =max [min (0.6,0.5),min (0.3,0.4)] = max (0.5, 0.3) = 0.5 max [min (0.6,0.3),min (0.3,0.7)] MT(x1,z3)=max [min (0.6,0.3),min (0.3,0.7)] = max [0.3, 0.3] = 0.3 MT(x2,z1)=max [min (0.2,1),min (0.9,0.8)] MT(x2,z1)=max [min (0.2,1),min (0.9,0.8)] = max [ 0.2, 0.8] = 0.8 =max [min (0.2,0.5),min (0.3,0.4)] MT(x2,z2)=max [min (0.2,0.5),min (0.3,0.4)] = max [0.2, 0.4] = 0.4 max [min (0.2,0.3),min (0.9,0.7)] MT(x2,z3)=max [min (0.2,0.3),min (0.9,0.7)] = max (0.2, 0.7) = 0.7 T = RoS = [0.6 0.5 0.3 0.8 0.4 0.7]
  • 29. Example(Max-Prod) T = R . S MT(x1,z1) = max [MR(x1,y1).MS(y1,z1)] = max [MR(x1,y1).MS(y1,z1)] MR(x1,y2).MS(y2,z1)] MR(x1,y2).MS(y2,z1)]= max (0.6, 0.24) = 0.6 MT(x1,z2)=max [MR(x1,y1).MS(y1,z2)] =max [MR(x1,y1).MS(y1,z2)] [MR(x1,y2).MS(y2,z2)][MR(x1,y2).MS(y2,z2)] = max [0.30, 0.12] = 0.30
  • 30. Example(Max-Prod) MT(x1,z3)=max [0.18,0.21]=0.21 MT(x2,z1)=max [0.2,0.72]=0.72 MT(x2,z2)=max [0.10,0.36]=0.36 MT(x2,z3)=max [0.06,0.63]=0.63 z1 z2 z3 T= R.S = x1 [ 0.6 0.3 0.21 ] x2 [ 0.72 0.36 0.63]