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Fuzzy Logic IMportantLecture Notes Notes book V.V.IMP
Master in Science in IT
Ai And Soft Computing (University of Mumbai)
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1.2
FUzZY LOGIC
Fuzzy set theory proposed in 1965 by Lotfi A. Zadeh (1965) is a generalization of classical set
theory. Fuzzy Logic representations founded on Fuzzy set theory try to capture the way humans
represent and reason with real-world knowledge in the face of uncetainty. Uncertainty could arise
due to generality, vagueness, ambiguity, chance, or incomplete knowledge.
A fuzzy set can be defined mathematically by assigning to
universe of discourse, a value representing
in
each
possible
individual
the
its grade of membership in the fuzzy set. This grade
with the concept
corresponds to the degree to which that individual is similar or compatible
represented by the fuzzy set. In other words, fuzzy sets support a flexible sense of membershipof
elements to a set.
to a set and hence,
theory, an element either belongs to or does not belong
(between 0 and I)
such sets are termed crisp sets. But in a fuzzy set, many degrees of membership
associated with a fuzzy set A such that the
are allowed. Thus, a membership function MA(x) is
the interval [0, 1].
function maps every element of the universe of discourse X to
universe of discourse), the fuzzy set "tall"
For example, for a set of students in a class (the
to
are tall with a degree of membership equal
(fuzzy set A) has as its members students who
of membership equal to 0.75 (4aC)
are of medium height with a degree
who
students
1),
1(4Ax)
to 0 (MAx)
0), to cite a few
are dwarfs with a degree of membership equal
who
those
and
= 0.75)
values between
student of the class could be graded to hold membership
cases. In this way, every
on their height.
0 and 1 in the fuzzy set A, depending
sets to express gradual transitions from membership
The capability of fuzzy
= 0) and vice versa has a broad utility. It not only
(4A(T)
non-membership
to
<
1)
(0 HA(x)
measurement of uncertainties, but also
In classical set
=
=
of
meaningful and powerful representation
of vague concepts expressed in natural language.
a meaningful representation
difference, and
as union, intersection, subsethood, product, equality,
such
Operations
sets to varying degree of
on fuzzy sets. Fuzy relations associate crisp
defined
also
are
disjunction
of
such as union, intersection, subsethood, and composition
membership and support operations
provides for
provides for
a
relations.
Just
as
fuzzy logic.
crisp
rise
theory has influenced symbolic logic, fuzzy set theory has given
in symbolic logic, truth values True or False alone are accorded
set
While
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propositions, in fuzzy logic multivalued truth values such as true, absolutely true, fairky true,
rules (which are
false, absolutely false, partly false, and so forth are supported. Fuzzy inference
rule based systems
computational procedures used for evaluating linguistic descriptions) and fuzzy
in real-world problems.
(which are a set of fuzzy F-THEN rules) have found wide applications
especially promoted by
Fuzzy logic has found extensive patronage in consumer products
control systems, pattern recognition
the Japanese companies and have found wide use in
applications, and decision making, to name a few.
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Product
The
of two fuzzy
sets
product of two fuzzy
defined as
sets A and
B is
a new
fuzzy
set
A B whose membership function
H (x) H(r) 4;()
6.29)
=
Example
= {71, 0.2), (r2, 0.8). (xz, 0.4)}
B
Since
A
=
max = Unions
Min = Intersection
{ , 0.4) (2, 0), (rz, 0.1)}
B = {(71, 0.08) (p, 0) (3, 0.04)}
Hi.B(a)
=
i a)"z(az)
0.2
HaB(2)
=
0.4
0.08
Hj(z)Hz (z2)
0
is
0=
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=
0.8
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HaB(x3) = H(x3)Hz()
= 0.4 0.1
= 0.04
Equality
Two
fuzzy
sets
and
B are
said
to
be
equal (A
=
Ë)
if
uz(x)
=
H(t)
Example
{,0.2)(*2,0.8)}
B =((.0.6)(*,,0.8)
C ={(.0.2)(x,,0.8)}
Since
Pa()
Hz(x) although
but
Since
Ha(y) = HE (%) = 0.2
and
Product of a
H x2) = H(x2) = 0.8
fuzzy
Multiplying a fuzzy
set with
membership function
set A
a
by
crisp number
a
crisp number
a
results in
a new
fuzzy
set
P a ) = a-Hjt)
Example
A=
For
{(z,0.4), (z2,0.6), (*z,0.8)}
a = 0.3
{(.0.12), (rz 0.18), (x3.0.24)}
Ha)= a Hz(x1)
a
since,
=
= 0.3 0.4
0.12
PaA2)= a Hj(x2)
0.3 0.6
0.18
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product
a.
A with
t:
(6.3
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Haa)
aPj(x3)
= 0.3 0.8
= 0.24
Power of
a Juzzy
set
n a w e r of a fuzzy set A 1S a new fuzzy set A" whose membership function is given by
HA (x) = (4z («)"
n.idine
fuzzy
a
Is called
set to its
Dilation
(6.32)
second power is called Concentration
(CON) and taking the square
root
(DIL).
Example
{(x.0.4), (x2.0.2), (x3.0.7)}
a= 2
For
(A = {(.0.16), (F2,0.04). (x3,0.49
Hence
Ha:()
Since
=
(4z(z))* (0.4
=
=
Ha(2)= (4z(x)* (0.2
(Hz73)) (0.7)
H(a3)
=
0.16
=
=
0.04
=
=
0.49
Difference
The
uerence of two fuzzy sets
and B is a new fuzzy set A -B defined as
A-B=(
Example
A
(6.33)
{x,0.2), (rz,0.5). (x3,0.6)); B= {(%.0.1), (rz 0.4), (73,0.5))
B = x.0.9), (x2.0.6), (x3,0.5)}
=
{(x,0.2)(7z,0.5)(x.0.5))
Disjunctive sum
disjunctive sur
Sum
of two fuzzy
sets
and B is
a
new
fuzzy
set A
B defined
as
(6.34)
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176 Neural Netoorks, Fuzzy Logic, and GeneticAigorithms-ymthesis and Applications
Example
{.0.4)(x,,0.8)(x,0.6)
B = (z,0.2)(x,.0.6)(,0.9)1
A
Now,
= {lz,.0.6)%*,,0.2)(x3,0.4))
B°
{(,0.8)(r2,0.4)(x3,0.1)}
AnB = {(7,,0.2)(z,.0.2)(x3,0.4)}
An
={G;,0.4)(7g,0.4)(a,0.))
AB
6.3.3
=
(7,0.4)(T2,0.4) x3,0.4)}
Properties of Fuzzy Sets
Fuzzy sets follow some of the properties satisfied by crisp sets. In fact, crisp sets can be thoui
of as special instances of fuzzy sets. Any fuzzy set
membership of any element belonging to the null set
belonging to the reference set is 1.
is a subset of the reference set X. Also, te
is 0 and the membership of any elkemet
The properties satisfied by fuzzy sets are
Commutativity:
(6.39
Associativity:
U ) =(U B) u
An (Bn ) =(Ã n B) n
Distributivitry:
Au(Bn ) = (
B) n(ãu )
An(Bu ) =(Ä n B)udn )
Idempotence:
An
Identity:
Transitivitry: If
Involution:
De Morgan's laws:
AUX= X
C , then s
AY =
An By=(Au B°)
AU B = (Ä n B°)
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6.5.2 Operations
Let
on
Fuzzy Relations
and S be fuzzy relations
on Xx Y.
Union
HRUF(x,y)) = max (j4g(,). 4z(x,y))
(6.54)
Intersection
HRos(x,)
=
min(4^l(x,), Ag(x,y)
Complement
Hre (x,y) =l-Hr(zy)
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(6.56)
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6.5
FUZZY RELATIONS
fuzzy
n-tuples (1, X2,.., Xn)
Fuzzy relation is
a
set
defined
on
the Cartesian
product of crisp
X1, X... X whnere the
within
sets
of membership within the
may have varying degrees
of the relation between the tuples.
membership values indicate the strength
relatine
relation. The
Example
Let R be the fuzzy relation between two sets X1 and X2 where X1 is the set of diseases and
X
is the set of symptoms.
X
X2
The
=
{typhoid,
viral fever,
=
{running
nose,
fuzzy
6.5.1
high
common
cold}
temperature,
relation R may be defined
shivering}
as
Running
High
Shivering
nose
0.8
Typhoid
0.1
temperature
0.9
Viral fever
0.2
0.9
0.7
Common cold
0.9
0.4
0.6
Fuzzy Cartesian Product
Let A be
fuzzy
o n the universe.
defined on the universe X and B be a fuzzy set defined on the u
the Cartesian
product between the fuzzy sets A and B indicated as AxB and resu
fuzzy relation R is given by
a
set
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Fuzzy Set Theory
where
R has its
here R
R=AxBc XxY
given by
HRx,) = HixBl7,y)
Example
=
(652)
membership function
=
Let A
183
min
(6.53)
((x), Hg (y)
{(X1. 0.2), (T2, 0.7), (x3, 0.4)} and
=
sets defined
((V. 0.5). (ya. 0.6)) be two
the universes of discourse X = {x1, x2, xal and Y= {y1, yat respectively. Then fuzzy
the fuzzy relation on
r
resulting
out of the
fuzzy Cartesian product
A xB is
given by
y2
0.2
0.2
R AxB= *2 0.5 0.6
30.4 0.4
since,
) min ("z(x),Hz(y)
R(1.2)=min(0.2,0.6) =0.2
R(
=
R(x2,y) =min(0.7,0.5)
=
0.5
min (0.7,0.6)
=
0.6
=
0.4
R(2,y2)
=
R(x3.y)= min(0.4,0.5)
R(x3, y2) min (0.4,0.6)
=
=
=
min (0.2,0.5)
0.4
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=
0.2