fuzzy complement

1. FUZZY COMPLEMENT

2. CONTENTS
• INTRODUCTION
• HISTORY
• DEFINITION
• CRISP SET
• FUZZY COMPLEMENT
• FUZZY UNION
• FUZZY INTERSECTION
• APPLICATIONS
• REFERENCES

3. INTRODUCTION
Fuzzy set is a mathematical model of vague qualitative or quantitative data,
frequency generated by means of the natural language. The model is based on the
generalization of the classical concepts of set and its characteristic function. In
mathematics, fuzzy sets are somewhat like sets whose elements have degrees of
membership. In classical set theory, the membership of elements in a set is
assessed in binary terms according to bivalent condition – an element either
belongs or does not belong to the set. By contrast, fuzzy set theory permits the
gradual assessment of the membership of elements in a set; this is described with
the aid of membership function valued in the real unit interval [0,1]. Fuzzy sets
generalize classical sets, since the indicator functions of classical sets are special
cases of the membership functions of fuzzy sets, if the latter only take values 0 or
1. In fuzzy set theory, classical bivalent sets are usually called crisps sets. The fuzzy
set theory can be used in a wide range of domains in which information is
incomplete or imprecise, such as bioinformatics.

4. HISTORY
The concept of fuzzy set was published in 1965 by Lotfi A. Zadeh. Since that
seminal publication, the fuzzy set theory is widely studied and extended. Its
application to the control theory became successful and revolutionary especially
in seventies and eighties, the application to data analysis, artificial intelligence,
and computational intelligence are intensively developed, especially, since
nineties. The theory is also extended and generalized by means of the theory of
triangular norms and conorms, and aggregation operators. Fuzzy sets were
introduced independently by Lotfi A. Zadeh and Dieter Klaua in 1965 as an
extension of the classical notion of set. At the same time, Salii defined a more
general kind of structure called an L-relation, which he studied in an abstract
algebraic context. Fuzzy relations, which are used now in different areas, such as
linguistics, decision making, and clustering, are special cases of L-relations when
L is the unit interval [0,1].

5. DEFINITION:
Let X be a non-zero set. A fuzzy set A of this set X is defined by the following
set of pairs.
A={ ( x , 𝜇 𝐴(x) ) } : x ∈ X
Where, 𝜇 𝐴 : X → [ 0 , 1 ]
is a function called as membership function of A and 𝜇 𝐴 x is the grade of
membership or degree of belonginess or degree of membership of
x ∈ X in A.
Thus a fuzzy set is a set of pairs consisting of a particular element of the
universe and its degree of membership.
A can also be written as
A = {(𝑥1,𝜇 𝐴(𝑥1 )),(𝑥2, 𝜇 𝐴(𝑥2)),…….(𝑥 𝑛, 𝜇 𝐴(𝑥 𝑛))}

6. Symbolically we write
A =
𝑥1
𝜇 𝐴 𝑥1
,
𝑥2
𝜇 𝐴 𝑥2
,………
𝑥 𝑛
𝜇 𝐴 𝑥 𝑛
}
A fuzzy set has membership function that can take any value from 0 to 1.
A fuzzy set is said to be empty if its membership function 𝜇∅ is
identically zero.
A fuzzy set is universal if its membership function 𝜇 𝑥 is identically on X
i.e., 𝜇 𝑥(X) = 1.
Two fuzzy sets are said to be equal if 𝜇 𝐴(x) = 𝜇 𝐵(x) , ∀ x ∈ X.
A fuzzy set A(X) is said to be a subset of fuzzy set B(X),
𝜇 𝐴(x) ≤ 𝜇 𝐵(x) , ∀ x ∈ X.

7. CRISP SET:
A crisp set is a collection of well defined objects. The term well defined help us
to discriminate members and non-members of the cell.
Properties of crisp set operations:

8. Support of a Fuzzy Set:
The support of a fuzzy set A is S(A) which is a crisp set ∀ x ∈ X, such that
𝜇 𝐴(x) > 0.
S(A) = {x ∈ X | 𝜇 𝐴(x) > 0}
The 𝛼 level set OR 𝛼 cut:
The α level set is a crisp set of elements that belongs to the fuzzy set A at
least to the degree α i.e.,
𝐴 𝛼 = {x ∈ X | 𝜇 𝐴(x) ≥ α}
Strong 𝜶 level set:
The strong α level set is defined as
𝐴 𝛼
′ = {x ∈ X | 𝜇 𝐴(x) > α}

9. Normalised Fuzzy Set:
A fuzzy set is called normalised when its element attains the maximum
possible membership grade.
Characteristic equation:
The characteristic function 𝜇 𝐴 : X → { 0 , 1 }, such that
𝜇 𝐴(x) = 1 , x ∈ A
𝜇 𝐴(x) = 0 , x ∉ A
It assign value 1 to a member and 0 to a non-member. Hence, it is also called
as membership function.
Cardinality:
The number of elements of a set is called the cardinality of the set and is
denoted by 𝐴 .

10. FUZZY COMPLEMENT
Compliment of a fuzzy set A is the set of elements of X excluding those of set A
𝐴 = {x | x ∉ A}
In membership function:
𝜇 𝐴(x) = {x ∈ X | 1- 𝜇 𝐴(x)}
A complement of a fuzzy set A is specified by a function
c : [0,1] → [0,1],
which assigns a value c(𝜇 𝐴(x)) to each membership grade 𝜇 𝐴(x). This assigned
value is interpreted as the membership grade of the element x in the fuzzy set
representing the negation of the concept represented by A. Thus, if A is the
fuzzy set of tall men, its complement is the fuzzy set of men who are not tall.
Obviously, there are many elements that can have some nonzero degree of
membership in both a fuzzy set and its complement.

11. FUZZY UNION
The union of fuzzy set A and fuzzy set B is a fuzzy set C, which
is given as
C = A∪B
Where, 𝜇 𝐶(x) = 𝜇 𝐴(x) ∪ 𝜇 𝐵(x)
𝜇A∪B(x) = max {𝜇 𝐴(x) ; 𝜇 𝐵(x)}

12. FUZZY INTERSECTION
The intersection of fuzzy set A and fuzzy set B is a fuzzy set C
which is given as
C = A∩B
Where, 𝜇 𝐶(x) = 𝜇 𝐴(x) ∩ 𝜇 𝐵(x)
𝜇A∩B(x) = min {𝜇 𝐴(x) ; 𝜇 𝐵(x)}

13. APPLICATIONS
1. Traffic monitoring system
2. Applications in commercial appliances like AC and heating
ventilation.
3. Gene expression data analysis.
4. Facial pattern recognition.
5. Weather forecasting systems.
6. Transmission systems.

14. 7. Antiskid braking system.
8. Control of subway systems and unmanned helicopters.
9. Knowledge based systems for multiobjective
optimization of power systems.
10. Models for new product pricing or project risk
assessment.
11. Medical diagnosis and treatment plans.

15. THANK YOU