Fuzzy sets

INTRODUCTION TO FUZZY SETS
S. Anita Shanthi
Department of Mathematics,
Annamalai University,
Annamalainagar-608002,
Tamilnadu, India.
E-mail : shanthi.anita@yahoo.com
S. Anita Shanthi (Department of Mathematics, Annamalai University, Annamalainagar-608002,Tamilnadu, India. E-mail : shanthi.anita@INTRODUCTION TO FUZZY SETS 1 / 43

Introduction
Introduction
The theory of fuzzy set was first introduced by Lotfi A.Zadeh in
1965. Fuzzy set theory has been developed in many directions by
many scholars and has evoked great interest among mathematicians
working in different fields.

Preliminaries
Preliminaries
In this section we give some useful deﬁnitions with examples.
Deﬁnition 1.
The Universe of Discourse is the range of all possible values for an
input to a fuzzy system.

Preliminaries
Deﬁnition 2.
Let X denote a universal set. Then, the membership function µA by
which a fuzzy set A is usually deﬁned is of the form
µA : X → [0, 1]
where [0, 1] denotes the interval of real numbers from 0 to 1.

Preliminaries
Deﬁnition 3.
A fuzzy set, usually denoted by A is any set that allows its members
to have diﬀerent grades of membership (membership function) in the
interval [0,1].

Preliminaries
Example 4.
Consider the crisp universal set X of ages selected as:
X = {5, 10, 20, 30, 40, 50, 60, 70, 80}
and the fuzzy sets A-infant, B-adult, C-young and D-old as follows:

Preliminaries
EXAMPLES OF FUZZY SETS
X(ages) Infant Adult Young Old
5 0 0 1 0
10 0 0 1 0
20 0 .8 .8 .1
30 0 1 .5 .2
40 0 1 .2 .4
50 0 1 .1 .6
60 0 1 0 .8
70 0 1 0 1
80 0 1 0 1

Preliminaries
Deﬁnition 5.
The support of a fuzzy set A in the universal set X is the crisp set
that contains all the elements of X that have nonzero membership
grade in A.
supp(A) = {x ∈ X|µA(x) > 0}.

Preliminaries
Example 6.
The support of the fuzzy set young from the table is the crisp set
supp(young) = {5, 10, 20, 30, 40, 50}.

Preliminaries
Note.
An empty fuzzy set has an empty support; i.e., the membership
function assigns 0 to all elements of the universal set. The fuzzy set
infant is one example of an empty fuzzy set within the chosen
universe.

Preliminaries
Deﬁnition 7.
A fuzzy singleton is a fuzzy set whose support is a single point in X
with a membership function of one.

Preliminaries
Deﬁnition 8.
The crossover point of a fuzzy set is the element in X at which its
membership function is 0.5.

Preliminaries
Deﬁnition 9.
The height of a fuzzy set is the largest membership grade attained by
any element in that set.

Preliminaries
Deﬁnition 10.
A fuzzy set is called normalized when at least one of its elements
attains the maximum possible membership grade.

Preliminaries
Deﬁnition 11.
An α-cut of a fuzzy set A is a crisp set Aα that contains all the
elements of the universal set X that have a membership grade in A
greater than or equal to the speciﬁed value of α.
Aα = {x ∈ X|µA(x) ≥ α.

Preliminaries
Example 12.
For α = .2, the α-cut of the fuzzy set young from the table is the
crisp set
young.2 = {5, 10, 20, 30, 40}
For α = .8,
young.8 = {5, 10, 20}
For α = 1,
young1 = {5, 10}.

Preliminaries
Note.
The set of all α-cuts of any fuzzy set on X is a family of nested crisp
subsets of X.

Preliminaries
Deﬁnition 13.
The set of all levels α ∈ [0, 1] that represent distinct α-cuts of a
given fuzzy set A is called a level set of A.
ΛA = {α| µA(x) = α for some x ∈ X,
where ΛA denotes the level set of fuzzy set A deﬁned on X.

Preliminaries
Examplr 14.
Λyoung = {.2, .8, 1}.

Preliminaries
Definition 15.
The scalar cardinality of a fuzzy set A defined on a finite universal set
X is the summation of the membership grades of all the elements of
X in A. Thus,
|A| =
x∈X
µA(x).

Preliminaries
Example 16.
The scalar cardinality of the fuzzy set old from the table is
| old| = 0 + 0 + .1 + .2 + .4 + .6 + .8 + 1 + 1 = 4.1.

Preliminaries
Deﬁnition 17.
If the membership grade of each element of the universal set X in
fuzzy set A is less than or equal to its membership grade in fuzzy set
B, then A is called a subset of B. Thus, if
µA(x) ≤ µB(x), for every x ∈ X, then A ⊆ B.

Preliminaries
Example 18.
The fuzzy set old from the table is a subset of the fuzzy set adult
since for each element in our universal set
µold (x) ≤ µadult(x).

Preliminaries
Deﬁnition 19.
Fuzzy sets A and B are called equal if
µA(x) = µB(x) for every element x ∈ X.
This is denoted by A = B.

Preliminaries
Deﬁnition 20.
Fuzzy sets A and B are not equal if
µA(x) = µB(x) for at least one x ∈ X.
We write A = B.

Preliminaries
Example 21.
None of the four fuzzy sets deﬁned in the table is equal to any of the
others.

Preliminaries
Deﬁnition 22.
Fuzzy set A is called a proper subset of fuzzy set B when A is a
subset of B and the two sets are not equal. i.e.,
µA(x) ≤ µB(x) for every x ∈ X and
µA(x) < µB(x) for at least one x ∈ X.
We denote it as A ⊂ B if and only if A ⊆ B and A = B.

Preliminaries
Example 23.
The fuzzy set old from the table is a subset of the fuzzy set adult and
these two fuzzy sets are not equal. Therefore, old can be said to be a
proper subset of adult.

Operations on fuzzy sets
Deﬁnition 1.
The union of two fuzzy sets is deﬁned as
µA ∪ B(x) = max[µA(x), µB(x)].

Example 2.
When we take the union of fuzzy sets young and old from the table
the following fuzzy set is created.
young=
{(5, 1), (10, 1), (20, .8), (30, .5), (40, .2), (50, .1), (60, 0), (70, 0), (80, 0)}
old =
{5, 0), (10, 0), (20, .1), (30, .2), (40, .4), (50, .6), (60, .8), (70, 1), (80, 1)}
young ∪old =
{(5, 1), (10, 1), (20, .8), (30, .5), (40, .4), (50, .6), (60, .8), (70, 1), (80, 1)}

Deﬁnition 3.
The intersection of two fuzzy sets is a fuzzy set A ∩ B such that
µA ∩ B(x) = min[µA(x), µB(x)].

Example 4.
The intersection of fuzzy sets young and old is given below:
young=
{(5, 1), (10, 1), (20, .8), (30, .5), (40, .2), (50, .1), (60, 0), (70, 0), (80, 0)}
old =
{5, 0), (10, 0), (20, .1), (30, .2), (40, .4), (50, .6), (60, .8), (70, 1), (80, 1)}
young ∩old =
{(5, 0), (10, 0), (20, .1), (30, .2), (40, .2), (50, .1), (60, 0), (70, 0), (80, 0)}

Deﬁnition 5.
The complement of a fuzzy set with respect to the universal set X is
denoted by A and deﬁned by
µA(x) = 1 − µA(x), for every x ∈ X.

Example 6.
The fuzzy set not old is given by:
old =
{5, 0), (10, 0), (20, .1), (30, .2), (40, .4), (50, .6), (60, .8), (70, 1), (80, 1)}
not old=
{5, 1), (10, 1), (20, .9), (30, .8), (40, .6), (50, .4), (60, .2), (70, 0), (80, 0)}

Deﬁnition 7.
Matrix multiplication could be done using max-min composition and
max-product composition.

Example 8.
Consider the two matrices





.3 .5 .8
0 .7 1
.4 .6 .5





◦





.9 .5 .7 .7
.3 .2 0 .9
1 0 .5 .5





product = min and sum = max. The two matrices can be multiplied
using max-min composition as follows:

max(.3, .3, .8) = .8;max(.3, .2, 0) = .3;
max(.3, 0, .5) = .5;max(.3, .5, .5) = .5
max(0, .3, 1) = 1;max(0, .2, 0) = .2;
max(0, 0, .5) = .5;max(0, .7, .5) = .7
max(.4, .3, .5) = .5;max(.4, .2, 0) = .4;
max(.4, 0, .5) = .5;max(.4, .6, .5) = .6

The resultant matrix is





.8 .3 .5 .5
1 .2 .5 .7
.5 .4 .5 .6






Example 9.
Consider the two matrices





.3 .5 .8
0 .7 1
.4 .6 .5





⊙





.9 .5 .7 .7
.3 .2 0 .9
1 0 .5 .5





product = product and sum = max. The two matrices can be
multiplied using max-product composition as follows:

max(.27, .15, .8) = .8;max(.15, .1, 0) = .15;
max(.21, 0, .4) = .4;max(.21, .45, .4) = .45
max(0, .21, 1) = 1;max(0, .14, 0) = .14;
max(0, 0, .5) = .5;max(0, .63, .5) = .63
max(.36, .18, .5) = .5;max(.2, .12, 0) = .2;
max(.28, 0, .25) = .28;max(.28, .54, .25) = .54

The resultant matrix is





.8 .15 .4 .45
1 .14 .5 .63
.5 .2 .28 .54






References
References
[1] George J. Klir and Tina A. Folger, Fuzzy sets, uncertainity and
information.

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Fuzzy sets

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