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
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
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.
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