FUZZY SET THEORY SOME USEFUL DISCUSSIONS AND INVESTIGATIONSSurajit Bhattachaaryya ^{1}^{} ^{1} Department of Mathematics Seth Anandram Jaipuria College, Kolkata (The University of Calcutta), West Bengal, India 




Received 15 July 2021 Accepted 30 July 2021 Published 31 August 2021 Corresponding Author Surajit
Bhattachaaryya, surajit_bhattacharyya@yahoo.com DOI 10.29121/granthaalayah.v9.i8.2021.4124 Funding:
This
research received no specific grant from any funding agency in the public,
commercial, or notforprofit sectors. Copyright:
© 2021
The Author(s). This is an open access article distributed under the terms of
the Creative Commons Attribution License, which permits unrestricted use, distribution,
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credited. 
ABSTRACT 


In this paper I have discussed some basic but very
important theories of fuzzy set theory with numerous examples. I have
investigated αsets, operations of fuzzy numbers, on interval fuzzy sets
and also on fuzzy mappings. I have introduced S.Bs. class of fuzzy
complements with its increasing and decreasing generators. 


Keywords: The
αCut Sets, Fuzzy Unions, S.Bs.
Class of Fuzzy Complements, Fuzzy Numbers 1. INTRODUCTION In classical set
theory, an entity can either belong to a set or not; and in optimization, a
solution is either feasible or not. Precision assumes that the parameters
represent exactly either our perception of the phenomenon modeled or the features
of the real system that has been modeled. This is true for both the
deterministic and the stochastic cases. Thus, according to the traditional
view, mathematics must strive for certainty in all its manifestations (specificity,
precision, sharpness, feasibility, consistency, etc.). Hence, uncertainty
(nonspecificity, imprecision, vagueness, nonfeasibility, inconsistency
etc.) is regarded as unscientific. Now, we have a question, why do we need to study fuzzy set theory ? Let us take an example. If we assume the probability of an element to
be a member of a set is 80%, the final conclusion is still either it is or it
is not a member of the set. The chance for one to make a correct prediction
for an element as it is a member of the set is 80%, which does not mean that
it has 80% membership in the set and at the same time it possesses 20%
nonmembership. That is, there is an ambiguity or uncertainty about an
element to be a member or nonmember of the set. In the classical set theory, it is not
allowed an element to be a member of a set and not a member of the set at the
same time. Thus, many realworld problems cannot be described and handled by
the classical set theory. On the contrary, fuzzy set theory accepts partial
memberships and thus to some extent generalizes the classical set theory.
According to the modern view, uncertainty or fuzziness is considered essential
to science, it is not only an unavoidable plague, but it has a great utility. 


When we calculate the profit of a company, we can’t mention a particular percentage of profits as a modest profit. Its meaning is not totally arbitrary, however, we can’t use “modest profit” to mean precisely profit of 80% and neither, in fact, is a profit of 40%. If for, for any instance, a profit of 60% or more is considered modest, does it mean that a profit of 58% is not modest? This is clearly unacceptable, since, a difference of 2% profit hardly seems to be a distinguishable characteristic between the modest and not modest. But where do we draw the line? In order to resolve this paradox, the term modest may introduce vagueness by allowing some sorts of gradual transition from degree of profit that are considered to be modest and those are not. This is, in fact, precisely the basic concepts of fuzzy set, a concept that is both simple and intuitively pleasing and forms in a sense, a generalization of the classical or crisp set.
Fuzziness can be found in many areas of daily life, such as engineering, medicine, meteorology, manufacturing, management and so many. However, it is frequent in all areas in which human judgment, reasoning, learning, investigation, decision making are important. Generally, it is agreed that an important point in the evolution of the modern concept of uncertainty was the publication of a seminal paper by L. A. Zadeh [1965] .In this paper, Zadeh introduced a theory  Fuzzy sets  are sets with boundaries that are not precise .The membership in a fuzzy set is not a matter of affirmation or denial, but rather a matter of a degree.
When A is a fuzzy set and ‘a’ is one of its elements, the proposition “a is a member of A” is not necessarily either true or false, but it may be true only to some degree, the degree to which ‘a’ is actually a member of the fuzzy A.
In general, we deal with problems in terms of systems that are constructed as models of either some aspects of reality or some desirable manmade objects .The purpose of constructing models of the former type is to understand some phenomenon of reality, be it natural or manmade, making adequate predictions or retrodictions, learning how to control the phenomenon in any desirable way and utilizing all these capabilities for various ends. Models of later type are constructed for the purpose of prescribing operations. In constructing a model, we always attempt to maximize its usefulness. This aim is closely connected with the relationship among three key characteristics of every system model: complexity, credibility and uncertainty. Our challenge in system modeling is to develop methods by which an optimal level of allowable uncertainty can be estimated for each modeling problem because allowing more uncertainty tends to reduce complexity and increase credibility of the resulting model.
In a paper L. A. Zadeh (1973) wrote, “As the complexity of a system increases , our ability to make precise and yet significant statements about its behavior diminishes until a threshold is reached beyond which precision and significance ( or relevance) become almost mutually exclusive characteristics .” as we know that , a complete description of a real system often would want far more detailed data than a human being could ever recognize , simultaneously process and understand.
In the natural languages, the meaning of words is very often vague. The meaning of a word might be well defined, but when using the word as a label for a set, the boundaries within which objects do or do not belong to the set become fuzzy or vague. Examples of such words are “animals”, “birds”, “big trees”, “red flowers”, “tall men”, “good students”, “beautiful women”, “highly contagious diseases”, “numbers much greater than 10” and so many.
2. RECALL: ………………………………………
1) The
support of a fuzzy set Ā within a universal set U is the crisp set that contains
all elements of U that
have nonzero membership grades in Ā
2) The height of a fuzzy set is the largest membership grade obtained by any element in that set.
3) Hausdorff distance h (X, Y) = max
is symmetric.
4) Every fuzzy complement has at most one equilibrium.
5) Let A = [A (i, j)] be an interval matrix. Then the real nonnegative matrix w (A) = [ w (A (i, j))] is called the width of A.
6) For interval matrices A, B, C, D we have the inclusion monotonic property like…
A C and B D ⇒ A B C D.
7) The law of contradiction and law of excluded middle are violated for the fuzzy sets.
8) Every fuzzy number is a convex fuzzy set .
3. CRISP SETS
A classical or crisp set is generally defined as a collection of objects x U, universal set. Each single element can either belong to or not belong to a set A, A U. In the first case, the statement “x belongs to A” is true, while in the second case this statement is false.
We can define a crisp set in various ways: either by
enumerating the elements that belong to the set; describing the set analytically
like, (A = {x │x ≤ 10 and a
positive integer}) i.e. setbuilder form or defining the elements by using the
characteristic function, in which 1 indicates membership and 0 indicates
nonmembership.
A set can be defined by a function, usually called a characteristic function, that declares which elements of U are members of that set and which are not. Set A is defined by its characteristic function ^{[1]} χ _{A}, as follows ……..
This characteristic function maps elements of X to elements of the set {0, 1}, which is usually expressed by
χ _{A }: U → {0, 1}.
For each for x U , when χ _{A} (x ) = 1 , x is declared to be a member of A and when χ _{A} (x ) = 0 , x is declared a nonmember of A.
It is very much clear from the definition that a sharp,
unambiguous
distinction exists between the members and the nonmembers of the set.
4. FUZZY SET
A fuzzy set can be defined by assigning to each possible
individual in the universe of discourse a value representing its grade of
membership in the fuzzy set. This grade corresponds to the degree to which that
individual is similar with the concept represented by the fuzzy set. Thus,
individuals may lie in the fuzzy set to a greater or lesser degree as indicated
by a larger or smaller membership grade.
The characteristic function ^{[1]}, defined earlier, can
be generalized in such a manner that values assigned to the elements of the
universal set fall within a specified range of unit interval [0, 1] and
indicate the membership grade of these elements.
If U is a collection
of objects denoted by x, then a fuzzy
set Ā in U is a set of
ordered pair …..
Ā = {( x , μ _{Ā}( x ))
│x U}
μ
_{Ā }( x ) is called the membership function or grade of
membership i.e. μ _{Ā } : U→ [0, 1]. In this case, each membership function maps elements of an
universal set U
into
real numbers in [0, 1]. Each fuzzy set is
completely
uniquely defined by one particular membership function; so the symbols of membership functions may also be used
as labels of associated fuzzy sets.
In this text, we will use Ā as a fuzzy set and μ _{Ā
} as its membership function.
Example:4.1. ^{[2]} Let us
consider U =
{u,
v, w, x, y, z } and μ _{Ā}_{ } : U→ [0, 1] defined by
μ _{Ā }(u) = 0, μ _{Ā}(v) =
0.2, μ _{Ā}(w)
= 0.6, μ _{Ā}(x) = 0.8,_{ }μ _{Ā}(y)
= 0.45 and _{ }μ _{Ā}(z)
= 1._{ } Then the fuzzy set Ā can be written as :
Ā = { (u, 0 ), (v, 0.2 ), (w,0.6),
(x, 0.8), (y, 0.45), (z, 1) }.
Example:4.2. Let us consider a
fuzzy set Ā = “real numbers close to 5” (see Figure 1)
Here Ā = {(
x ,
μ _{Ā}( x ))│x U},U is the universal set, where the
membership function is μ
_{Ā}( x ) = 1 + (x ̶ 5^{2})^{—1}.

Figure 1 
Thus, a fuzzy set is denoted by an ordered pairs, the first
element of which denotes the element and the second the degree of membership.
Example: 4.3. Let U be the
set of all human beings i.e. universal set and let Ā = {( x U
│x is weighty } . Then Ā is a “fuzzy set” because the property
“weighty” is not well defined and can’t be precisely measured as for a person
who weighs 58 kg., it is not clear whether he belongs to the set Ā or not.
For the time being, we may describe the concept “weighty” person
by the curve shown in Figure 2, using common sense, where the only persons who are considered to
be absolute “weighty” are those, whose weights are 120 kg. or more and the
persons who are considered to be not absolute “weighty”
or of “zero weight” are those new borns (approx.). For example, a person with
50 kg. of weight is considered to be “weighty” with “degree 0.5” and at the same time also not
“weighty” with “degree 0.5” according to the measuring curve we have used in Figure 2. We can’t
exclude this person from the set Ā , nor include him completely
.
Moreover, the two concepts are conflicting; x is
“weighty” or not “weighty” at the same time, which classcal
mathematics can’t accept. Such a vague and conflicting
description of a set is acceptable by
fuzzy mathematics, however, which turns out to be very useful in many
realworld applications.
Here, the curve, introduced in Figure 2, can be used to
define the partial membership of weight of
any person related to set Ā. Here, Ā
= {( x ,
μ _{Ā}( x )) │x U}
Where the membership function is μ _{Ā}( x
) = 0 , if x ≤ 0
1 ̶ e^{ }^{̶ }^{ }^{x}^{
} , if x
˃ 0

Figure 2 
Example: 4.4.
Let us now consider, as another simple
example, three fuzzy sets defined within a finite universal set that consists
of eight levels of effects of CORONA VIRUSES on human beings.
0 – No symptoms
1 – Mild fever and cough.
2 – Tiredness, tastelessness and loss of smell.
3 – Home isolation is required with proper medication.
4 – Difficulties in breathing , throat problem.
5 – Hospitalization is required.
6 – Under ventilation.
7 – Death.
Membership functions of three fuzzy sets, which attempt to capture
the concepts of littleeffected, highly effected, very seriously effected
persons are defined in Figure 3. by three colors.

Figure 3 
Example: 4.5.
Several fuzzy sets representing the concepts like very low, low, medium,
high, very high and so on are often used to define different states of a
variable. Such a variable is generally called a fuzzy variable. In Figure 4, number of passengers, in a compartment of a
train, within a range [P_{1}, P_{2}] is characterized as a
fuzzy variable. They are all defined by the membership functions of the form…..
[P_{1}, P_{2}] → [0, 1]. Graphs of these functions have trapezoidal
shapes.

Figure 4 
The significance of fuzzy variables is that they facilitate
gradual transitions between states and possesses a natural capability to
express and deal with observation and measurement uncertainties, but its
usefulness depends critically on our capability to construct appropriate
membership functions for various given concepts in various contexts.
5. SOME BASIC CONCEPTS OF FUZZY SETS
Now, we want to discuss some basic concepts of fuzzy sets. To
illustrate the concepts, we consider three fuzzy sets that represent the
concepts of a weak, mediocre and good students in a class. A reasonable
mathematical expression of these concepts by trapezoidal membership functions
μ _{Ā1}, μ _{Ā2 }and μ _{Ā3 }is shown in Figure 5. These functions are defined on the the
interval [0, 100] as follows:
Here, by x we denote % of
marks obtained by a student.

Figure 5 
Definition (1): The two subsets ^{α} Ā and ^{α+} Ā , of
Ā defined by
^{α} Ā = {x ϵ Ā │ μ _{Ā
}(x) ≥ α}
α ϵ (0, 1]
and ^{α+} Ā
= {x ϵ Ā │ μ _{Ā }(x) ˃ α} α ϵ [0, 1)
are called “αcut set” or “αlevel set” and the “strong
αcut set” or “strong αlevel set”, respectively. The corresponding
characteristic function of αlevel set can be represented as follows:
Example: 5.1. Now, we refer to Example:4.1^{[2]}
and try to make a list of possible αcut sets:
^{.2} Ā
= {v, w, x, y, z }
^{.45}Ā = {w, x, y, z
}
^{.6} Ā
= {w, x, z }
^{ 1} Ā = {z}
The strong αcut set for α = .8 is ^{.}^{8+} Ā = { z }.
Definition(2): A fuzzy set is convex
iff its every αcut subset is convex i.e. for any x_{1,} x_{2 } U and any λ ϵ [0, 1 ] we have
μ _{Ā }( λx_{1 }+
(1—λ)x_{2} ) ≥ min.{ μ _{Ā}( x_{1}
), μ _{Ā}( x_{2} ) } .
It must be very clear that the definition of convexity for fuzzy
sets does not mean that the membership function of a convex fuzzy set is a
convex function. Actually, those are concave and not convex, according to
standard definitions.
Theorem 5.1: A “strong αcut
subset” of a fuzzy set Ā is convex if its “αcut
subset” is convex.
Proof: Proof is very
obvious from definition.
Definition (3): Union and intersection of two fuzzy sets:
Given two fuzzy
sets Ā and Ḃ , their standard union Ō = Ā Ḃ is pointwise defined by
μ _{Ō }( x ) =
max { μ _{Ā}( x_{ }), μ _{Ḃ}( x ) }
, x ϵ U.
Given two fuzzy
sets Ā and B , their standard intersection Ē = Ā B is pointwise defined by
μ _{Ē }( x ) =
min { μ _{Ā}( x_{ }), μ _{Ḃ}( x ) }
, x ϵ U.
where max and min are the maximum and minimum
operator, respectively.
Definition(4): Complement of a set and the equilibrium point:
The membership function of the standard
complement cĀ of a fuzzy set Ā
, is defined by
μ _{cĀ} ( x ) = 1 μ_{Ā} ( x ) , x ϵ
U. [ In this text, we will use cĀ
as a complement set of a fuzzy set Ā.]
The equilibrium of a complement c is that degree
of membership in a fuzzy set Ā which is equal to the degree of membership
in complement cĀ i.e. which is a solution of the equation 1—x = x , x ϵ μ_{Ā} ( x ) .
Properties ^{[3]} of fuzzy complements: Function c, the
fuzzy complement may satisfy following axiomatic properties:
Axiom 5.1: If a ≤ b, then c(a) =
c(b), a, b ϵ [ 0, 1]. (Monotonicity)
Axiom 5.2: c (0) = 1, c (1) = 0.
(Boundary conditions)
Axiom 5.3: c is a continuous function.
Axiom 5.4: c is involutive i.e. c(c(x)) = x, x ϵ [ 0, 1].
Definition(5) : Cartesian product :
The cartesian product of n fuzzy sets Ā_{1},
… ….
……. …… ,Ā_{n }in U_{1}
, ….. ….. …… ,U_{n } is a fuzzy set in the product space U_{1} x .. ..
.. .. x U_{n } with the membership function
μ _{(Ā}_{1}_{ x … x Ā}_{n}_{)} ( x ) = min { μ _{Āi}( x_{i} ) │ x = (x_{1} , x_{2}………. x_{n }) , x_{i }ϵ U_{i} }_{}
_{ }_{ }^{i}
Example:5.2. Now, we refer to Example:4.1^{[2] }where
Ā
= { (u, 0 ), (v, 0.2 ), (w,0.6), (x, 0.8), (y,
0.45), (z, 1) }.
And assume = {(w, 0.2), (x,
0.4), (y, 0.48), (z, 0.9), (m, 1), (n, 1).
Then union , Ō = {(u, 0), (v,
0.2), (w, 0.6), (x, 0.8), (y, 0.48), (z, 1), (m, 1), (n, 1) }.
Intersection , Ē = {(w, 0.2), (x, 0.4), (y, 0.45), (z, 0.9) }
Complement, c = {(u, 1), (v, 1), (w, 0.8), (x, 0.6),(y,
0.72), (z, 0.1)}.
Example:5.3. Let Ā(x) = { (a, 0.3), (b, 1), (c, 0.8), (d, 0.6)}
and (x) = { (a, 1), (c, 0.6) }
Then Ā x = { [(a; a), 0.3], [(b; a),1],[(c; a),
0.8],[(d; a), 0.6],[(a; c), 0.3],[(b; c), 0.6],[(c; c),0.6],[(d; c),0.6] }.
where
a, b, c, d ϵ U.
Definition(6) : Let G be a group. A fuzzy subset Ā of the group G is called a
fuzzy subgroup of the group G if
μ _{Ā}( x y_{ }) ≥ min{ μ _{Ā}( x_{ }), μ _{Ā}( y_{ }) } , x, y ϵ G .
μ _{Ā}( x ^{1}_{ }) = μ _{Ā}( x_{ }) , x ϵ G
Definition(7): Fuzzy union of two fuzzy sets Ā_{1} and Ā_{2 }:
Let Ā_{1} be a fuzzy subset of a set X_{1}
and Ā_{2} be a fuzzy subset of a set X_{2}. Then the fuzzy
union of the fuzzy sets Ā_{1} and Ā_{2}
is defined as follows :…..
μ _{Ā1}μ _{Ā2} : X_{1
} X_{2 } → [0, 1] given
by
Example:5.4. Let X_{1}
= { a, b, c, d, e } and X_{2} =
{ b, d, f, h, j } be two sets .
Let us define μ _{Ā1}:
X_{1} → [0, 1] by μ _{Ā1}(x) = 1 , if
x = a, b
0.7 , if
x = c
0.16 , if x = d, e.
And define μ _{Ā2}
: X_{2} → [0, 1 ] by μ
_{Ā2}(x) = 1 ,
if x = b, d.
0.7, if x = f
0.16, if x = h, j.
Now it is easy to calculate μ _{Ā1 } μ _{Ā2} as follows……………..
(μ _{Ā1} μ _{Ā2} ) (x) = 1 ,
if x = a, b, d
0.7 , if x = c, f
0.16 , if x = e, h, j .
Definition(8): Fuzzy union of three fuzzy sets Ā_{1}, Ā_{2} and Ā_{3:}
Let Ā_{1} be a fuzzy subset of a set X_{1}, Ā_{2}
be a fuzzy subset of a set X_{2} and Ā_{3} be a fuzzy
subset of a set X_{3}. Then the fuzzy union of the fuzzy sets Ā_{1},
Ā_{2} and Ā_{3} is defined as
follows :…..
μ _{Ā1}μ _{Ā2}μ _{Ā3}
: X_{1 } X_{2 } X_{3 } → [0, 1] given
by
Example:5.6. Let X_{1} = {
a, b, c, d, e } , X_{2} = { b, d, f, h, j } and X_{3} = {a, c, e, g, i } be three
sets .
Let us define μ _{Ā1} : X_{1}
→ [0, 1 ] by μ _{Ā1}(x) =
1 , if
x = a, b
0.7
, if
x = c
0.16 , if x = d, e.
and define
μ _{Ā2} : X_{2} → [0, 1 ] by μ
_{Ā2}(x) = 1 ,
if x = b, d.
0.7 , if x
= f
0.16 , if x = h, j .
and define μ _{Ā3} : X_{3}
→ [0, 1 ] by μ _{Ā2}(x)
= 1 ,
if x = a
0.7 , if x
= c, e.
0.16 , if x = g, i .
Now it is easy to calculate
μ _{Ā1} μ _{Ā2} μ _{Ā3} as follows ……………..
(μ _{Ā1} μ _{Ā2}μ _{Ā3})(x)
= 1 ,
if x = a, b, d
0.7 , if x = c, e, f
0.16 , if x = g, h, i, j .
Theorem 5.2 : ^{α+}(cĀ) = ^{(1—α)}(cĀ)
Proof: Let x ϵ ^{α+}(cĀ) , which implies that x ∉ ^{α+}(Ā)
Hence Ā(x) ≤ α , ⇒ 1 – Ā(x) ≥ 1— α i.e.
cĀ(x) ≥ 1— α ,
which means that x ϵ ^{1 }^{α}(cĀ) .
Consequently, ^{α+}(cĀ) ^{(1—α)}(cĀ)
Conversely, for any y ϵ
^{(1—α)}(cĀ) , we
have y ∉ ^{1  }^{α}(Ā)
Hence Ā(y) ˂ 1 – α and 1
– Ā(y) ˃ α i.e. cĀ(y) ˃ α , which means that y ϵ ^{α+}(cĀ) i.e. ^{(1—α)}(cĀ)
^{α+}(cĀ).
Therefore, ^{α+}(cĀ) = ^{(1—α)}(cĀ).
We now want to introduce a special fuzzy set, α(^{α}Ā)
, by defining as…
α(^{α}Ā) = { x ϵ
Ā │μ _{α (αĀ)}(x) = α Ʌ _{ }χ_{
}_{α Ā} (x)
}_{ }, χ_{
}_{α Ā} (x)
is defined by equation … “( 1)”.
Theorem 5.3: α(^{α}Ā)(x) =
0.
Proof: α(^{α}Ā)(x)
= inf { α Ʌ _{ }χ_{
}_{α Ā} (x)
}_{ }
αϵ[0,1)
= inf { α Ʌ _{ }χ_{ }_{α Ā} (x) }_{ }V inf_{ }{ α Ʌ _{ }χ_{ }_{α Ā} (x) }_{ }
αϵ[0,μ _{Ā} ) αϵ[μ_{ Ā} ,1)
= _{ }inf { α Ʌ _{ }1 }_{ }V inf_{ }{ α Ʌ _{ }0 }_{ }
^{ }αϵ[0,μ
_{Ā}
) αϵ[μ _{Ā} ,1)
= _{ }inf { α }_{ }V 0_{}
αϵ[0,μ
_{Ā}^{ })
^{ }= 0. Hence proved. ^{ }
Theorem 5.4: ^{ } α(^{α+}Ā )(x) = 0.^{ }
Proof: Here, proof is similar to that of previous^{ }Theorem , so left for the readers.^{ }^{ }
Definition(9): Here we introduce a class of fuzzy
complements defined by ……………….
c_{λ}(a) = , where λ
ϵ
(2, ) , a ϵ [0, 1 ]
This class of functions
of fuzzy complements satisfies all the axioms given in properties ^{[3]}^{}. For different
values of parameter λ,
we obtain different involutive fuzzy complement. This class may be illustrated for
different values of λ as in Figure 6. For λ = 0, the function becomes the classical complement
i.e. a = 1—a, a ϵ [0, 1].^{ }Let us name this as the S.B’s.
class of fuzzy complements.
For any fuzzy complement c, the increasing generators are g(a) = a
where g is a continuous function from [0, 1] to ℝ such that g(0) =
0, g is strictly increasing and c(a) = g^{1}(g(1) – g(a)) a ϵ [0, 1 ].
For the S.Bs. class of fuzzy complements, the
increasing generators are ………….
g _{λ}(a)
= ln (1 + a) for if we take….
, a standard fuzzy complement
can be generated for λ = 0.
Interestingly, if we take a class of two parameter increasing
generators
g
_{λ,}_{
ω} (a) = ln (1 + a^{ω}) for λ
˃
 2 and ω ˃ 0 , we obtain a class of fuzzy
complements ………………..
c _{λ, ω }(a) = [ (1 – a^{ω }) / (1
+ a^{ω} ) ]^{1/ω}
which contains the S.B’s class ( for ω = 1) and the Yager class (
for λ = 0) as well.
Again, if there exists a continuous function f from [0, 1 ] to ℝ such that f(1) = 0, f is strictly decreasing and c(a) = f ^{1}(f(0) – f(a)) a ϵ [0, 1 ] , where c is a fuzzy
complement then f is a decreasing generator. Here
g(a)
= f(0) – f(a).
For the S.Bs. class, if
we take f(a) =  ln( ), then f is a decreasing function, f(1) = 0
and f satisfies the relation g(a) = f(0) – f(a).
[ since, f(0) =  ln( ) , so , 
ln( ) – { 
ln( ) } = ln (1 + a) = g(a). ]
Therefore, here f(a) =  ln( ) is a decreasing generator.

Figure 6 
6.
SOME ARITHMETIC
OPERATIONS ON INTERVALS
Here, we are concern about the situation where
the value of a member x of a set is uncertain. We assume, that the information
on the uncertain value of x provides an acceptable range a ≤ x ≤ b , where [ a, b ] R , is called the interval of confidence about
the value x . We mainly study closed and bounded intervals in this text, unless
otherwise stated.
Definition(10): Distance and width:
Let A = [a_{1} , a_{2} ] and B = [b_{1} , b_{2 }] be
intervals. Then distance between A and B is defined by
d(A,B) =
max { │a_{1}—b_{1}│ , │a_{2} – b_{2}│
}.
The width of an interval A = [a_{1} , a_{2} ] may be defined as w{A} = w{ [a_{1} , a_{2} ] } = a_{2} – a_{1} .
or equivalently^{[4]} w{A} = max
│x_{1} – x_{2}│
x_{1} ,x_{2}ϵ A
Theorem 6.1: Let A = [a_{1}
, a_{2} ] , B = [b_{1} , b_{2} ] , C = [c_{1} ,
c_{2} ] be the intervals. Then
i)
d(A + B , C + B ) = d(A , C ).
ii) d(A
– B, A – C ) = d(B, C).
iii) iii)│d(A,
C) – d(B, C)│≤ d(A, B ).
Proof:
i) d(A + B , C + B ) = max { │ (a_{1 }+_{ }b_{1})_{ }– (c_{1} + b_{1} ) │ , │ (a_{2 }+_{ }b_{2 })_{ }– (c_{2} + b_{2} ) │}_{}
= max {
│ (a_{1 }– c_{1}
│ , │ (a_{2 }– c_{2} │} = d (A , C ).
ii) d(A – B, A – C )
= max { │ (a_{1 }_{ }b_{1})_{ }– (a_{1}
 c_{1} ) │ , │ (a_{2
}_{ }b_{2 })_{
}– (a_{2}  c_{2} ) │} .
=
max { │ (c_{1 }_{ }b_{1}) │ , │ (c_{2 }_{
}b_{2 })_{ }│} = d(C , B) = d(B , C ).
iii) From
triangular law of addition, we have d(A,C) ≤ d(A,B) + d(B,C)
^{ } ^{ }
Theorem 6.2:^{ }Let A = [a_{1} , a_{2} ] , B = [b_{1} , b_{2} ] , C = [c_{1} , c_{2} ] be the intervals. Then d(AB , CB ) ≤ │B│ d(A , C)^{ }
^{ }
Proof: Here
, for convenience, we use l(A) = a_{1 }and u(A) = a_{2}
for any interval A = [a_{1} , a_{2} ] in this proof only.
Now, we have to prove that max {│l(AB) – l(CB)│, │u(AB) – u(CB)│} ≤ │B│ d(A , C)^{ }
First of all we are going to prove
│l(AB) – l(CB)│ ≤ │B│ d(A , C)^{ }
Without loss of generality, let us assume that l(AB)
≥
l
(CB),
Then since CB = {cb│c ϵ C, b ϵ
B } b ϵ B such that l(CB)
= l(Cb).
On the other hand, bA AB ⇒
that
l(bA)
≥
l(AB).
Hence , we
have l(bA)
 l(Cb) ≥ l(AB)
 l(CB) ≥ 0
So that, │ l(AB)
 l(CB)
│ = l(AB)
 l(CB)
≤ l(bA)
 l(Cb) = │l(bA)  l(Cb) │
≤
│b│d(A, C)
≤
│B│d(A, C).
In the similar manner, we can prove that │u(AB) –
u(CB)│} ≤ │B│ d(A , C).
Hence, we can conclude that d(AB , CB ) = max {│l(AB) – l(CB)│, │u(AB) – u(CB)│} ≤ │B│ d(A , C). ^{ }
Corollary: If the sets X, Y, Z, W ϵ U, (where U is the universal set)
Then │d(X, Y)—d(Z, W)│ ≤ d(X, Z) + d(Y, W).
Proof: │d(X, Y)—d(Z, W)│ ≤ │d(X, Y) – d(Y,
Z)│ + │d(Y, Z) –d(Z, W)│≤ d(X, Z) + d(Y, W).
Theorem 6.3: Let X and Y be any two intervals. Then……
(i) w{XY} ≥ w{X}│Y│
│X│w{Y} and w{XY} ≥ │X│w{Y}  w{X}│Y│.
(ii)
≤ + .
Proof:
In the similar manner we
can show the second inequality.
Hence, we can write that w{XY} ≥ │ w{X}│Y│
│X│w{Y} │
( iii) We have, = ≤
=
≤ + .
[ last step is due to the fact that , if x and y are positive, then ≤ = + ] ^{ }
Theorem 6.4: Let X = [x_{1}, x_{2} ] and Y = [y_{1}, y_{2} ] be two
intervals . Then X Y implies that…….
1) w{X} ≤ w{Y}
2) [w{X} + w{Y} ] ≤ d (X, Y ) ≤
[w{X} + w{Y} ]
Proof:
1)
Since X Y , we have y_{1 }≤ x_{1} ≤ x_{2} ≤ y_{2 }, from where it clearly follows that …
w{X}
= │x_{2} – x_{1}│
≤ │y_{2}
–y_{1}│ = w{Y}.
2) d (X, Y) = max{ │x_{1} – y_{1}│,
│x_{2} – y_{2}│}
= max{ (x_{1} – y_{1}), (x_{2} – y_{2})
}.
≤ y_{2} – x_{2} + x_{1} – y_{1}
= (y_{2
}– y_{1}) + (x_{1} – x_{2}) = │y_{2 }–
y_{1}│ + │x_{1} – x_{2}│
= │y_{2
}– y_{1}│ + │x_{2} – x_{1}│
= w{X} + w{Y}
…………………………………………………… ….. (5)
Again
d(X, Y ) = max{ │x_{1} – y_{1}│, │x_{2}
– y_{2}│} = max{ │x_{1}
– y_{1}│, │y_{2} – x_{2}│}
≥ (x_{1} – y_{1} + y_{2}
– x_{2}) = [(y_{2} – y_{1} ) + ( x_{1}
– x_{2})] = (│y_{2} – y_{1} │
+ │x_{1} – x_{2 }│)
= (│y_{2} – y_{1} │
+ │ x_{2} – x_{1 }│) = [w{X} + w{Y} ] ……………………………………………………… ……… (6)
From “(5)” and “(6)” it follows that … [w{X} + w{Y} ] ≤ d (X, Y ) ≤ [w{X} + w{Y} ]. ^{ }
^{ }
Example: 6.1. Let us take two
intervals X = [0.2, 0.5] and Y = [0.1, 0.6]. Here X Y.
We can verify the following values……
1)
w(X) = 0.3 , w(Y) = 0.5 ⇒ w{X} ≤ w{Y}
2)
X^{—1} = = [2, 5 ] , Y^{—1} = = [1.66, 10] ⇒ X^{—1} Y^{—1}.
3)
1 + X = [1.2, 1.5 ] and 1 +
Y = [1.1, 1.6 ] ⇒ 1 + X 1 + Y.
4)
1—X = [0.5, 0.8 ] and 1—Y = [0.4, 0.9 ] ⇒ 1 X 1 Y.
5)
= = [ ] = [0.67,
0.83], = = [0.62, 0.9 ] ⇒ .
6)
= [0.4, 0.9 ]
[0.62, 0.9] = [0.248, 0.81] and = [0.5, 0.8]
[0.67, 0.83] = [0.335, 0.664] ⇒ .
7.
OPERATIONS ON SOME INTERVAL MATRICES
7.1.
MULTIPLICATION
BETWEEN TWO INTERVAL MATRICES ^{[5]}
Here A is a matrix of
order 2x2 and B is a matrix of order 2x1. So, they are conformable to
multiplication and let A X B = C which is obviously of order 2x1.By the RULE OF
VACANCIES we can write
^{ }The position of first vacancy is 1^{st} row
and 1^{st} column. So this vacant position will be filled up by the
product of 1^{st} row of A with ^{ }1^{st} column of B, i.e. [2,
3]x[0,10] + [0, 2]x[6, 14] = [0, 30] + [0, 28] = [0, 58] . Similarly, the
second vacant position will be [1,3]x[0, 10] + [2, 3]x[6, 14] = [0, 30] + [12,
42] = [12, 72]
[ As , [m_{1}, n_{1}]x[m_{2},
n_{2}] = [p, q], where p = min(m_{1}m_{2}, m_{1}n_{2},
n_{1}m_{2}, n_{1}n_{2}) and q = max (m_{1}m_{2},
m_{1}n_{2}, n_{1}m_{2}, n_{1}n_{2}) ] .
C = [0,
58]
[12,72] .
7.2.
MULTIPLICATION
OF AN INTERVAL MATRICX BY A CONSTANT MATRIX
Let A = be an
interval matrix and be a constant
matrix.
We have to find out AB(here AB is possible).It will be
easy to find out the product , if before multiplication we write B as
Now we can easily
calculate AB as in the case of 7.1^{ [5]}.^{ }
So , AB
= [1, 1] [2, 0]
[0, 1] [0, 1]
8.
SOME OPERATIONS
ON FUZZY NUMBERS
Definition(11): Four basic arithmetic operations on fuzzy sets:
Let Ā and Ē be any two fuzzy numbers and the operation • ϵ { + ,  , . , / }. Then we can define a
fuzzy set Ā • Ē on ℝ , with the help of its αcut, in the form of …
Ā • Ē = α ^{α}(Ā • Ē ) where ^{α }(Ā • Ē ) = ^{α}Ā • ^{α}Ē^{ }
αϵ[0,1]
Definition(12): Alternatively ……
we can define a fuzzy set Ā • Ē on ℝ ,
with the help of an equation
(Ā • Ē)(s) = sup min[Ā(x), Ē(y)] , s ϵ ℝ
s = x•y
Definition(13): Fuzzy numbers MIN (Ā , Ē )and MAX(Ā , Ē )
If we want to write an
ordering of fuzzy numbers, first of all we have to extend the lattice
operations min and max on real numbers to the corresponding operations on fuzzy
numbers, MIN and MAX.For any fuzzy numbers A and B, we define…….
MIN (Ā , Ē )(s) = sup min[Ā(x), Ē(y)] and MAX (Ā , Ē) = sup min[Ā(x), Ē(y)]
s = min(x, y) s
= max(x, y)
we must remember that
every fuzzy number is a convex fuzzy set.
Example:8.1. Let
us consider two fuzzy numbers Ā = [4, 0] and Ē = [1, 5] , which can be
defined by their membership functions as follows …..
μ_{Ā} (x) = (x + 4)/ 2 , when
 4 ˂ x ≤ 2
(x)/2 ,
when 2 ˂
x ≤ 0
0 , when
x ≤  4 and x ˃ 0
μ_{Ē} (x ) =
(x1)/2 , when 1
˂ x ≤ 3
(5  x)/2 , when
3 ˂ x ≤ 5
0 otherwise
Calculate
Ā + Ē , Ā – Ē ,
Ē – Ā , Ā .
Ē , Ā / Ē .
According to
the given conditions ,^{α}Ā = [2α – 4,  2α ] and
^{α}Ē = [2α + 1, 5  2α ]
Then , for the resulting fuzzy numbers we can write_______
μ_{Ā + Ē} (x) = for  3 ˂ x ≤ 1.
, for
1 ˂ x ≤ 5
0 , elsewhere.
( Resulting membership
function of (Ā
+ Ē ) is illustrated in Figure 7)
μ_{Ā  Ē} (x) =
for  9 ˂ x ≤ 5.
, for
5 ˂ x ≤ 1
0 , elsewhere.
( Resulting membership
function of (Ā
– Ē) is illustrated in Figure 8)
μ_{ Ē Ā} (x) = for 1 ˂ x ≤ 5.
, for
5 ˂ x ≤ 9
0 , elsewhere.
( Resulting membership function of (Ē – Ā) is illustrated in Figure 9)
μ _{Ā }_{.}_{ Ē} (x) =
[ 4.5 –(0.25  x)^{1/2} ] /2 ,
for  20 ˂ x ≤  6.
[ 2.5  0.5(4x +
25)^{1/2 }] /2 , for
6 ˂ x ≤ 0
0 , elsewhere.
( Resulting membership
function of ( Ā . Ē ) is illustrated in Figure 10)
μ _{Ā }_{/}_{ Ē} (x) = for  4 ˂ x ≤  2/3.
, for
 2/3 ˂ x ≤ 0
0 , elsewhere.
( Resulting membership function of (Ā / Ē) is illustrated in Figure 11)

Figure 7 

Figure 8 

Figure 9 

Figure 10 

Figure 11 
Example:8.2. Calculate MIN(Ā , Ē) and MAX(Ā , Ē ) for the two fuzzy
numbers Ā = [ 6, 6] and Ē = [ 4,0] which can be defined as follows …..
.
μ_{Ā} (x) = (x + 6)/ 6 , when
 6 ˂ x ≤ 0
(6x)/6
, when 0 ˂ x ≤ 6
0
, when elsewhere .
μ_{Ē} (x ) =
(x + 4)/2 , when 
4 ˂ x ≤ 2
( x)/2 ,
when 2 ˂ x ≤ 0
0 otherwise

Figure 12 


Figure 13 

μ_{MIN(}_{Ā}_{ , }_{Ē}_{)}(x)
= 0
, when x ˂  6 and x ˃ 0 ,
illustrated in Figure 12
(x +
6)/ 6 , when  6 ˂ x ≤  3,
(x
+ 4)/2 , when 
3 ˂ x ≤ 2
( x)/2
, when 2
˂ x ≤ 0
μ_{MAX(}_{Ā}_{ , }_{Ē}_{ }_{)}(x) =
0 , when x ˂  4 and x ˃ 6 ,
illustrated in Figure 13
(x + 4)/2
, when  4 ˂ x ≤ 3
(x + 6)/ 6 ,
when  3 ˂ x ≤ 0
(6x)/6 ,
when 0
˂ x ≤ 6
9.
FUZZY MAPPING
Let S be a nonempty
set in R^{n} (denoted by E), a mapping f : S → F_{0} is
said to be a fuzzy mapping, where F_{0 }is the set of all fuzzy numbers.
Definition(14): ^{[6] }The lowerlevel set of f, denoted by S_{u}(f),
is a subset of E defined by
S_{u}(f) = { x
│ x ϵ S and f(x) ≤ u}, where
(u ϵ F_{0}).
The upperlevel set of
f, denoted by S^{u}(f), is a subset of E defined by
S^{U}(f) = {x
│ x ϵ S and f(x) ≥ u}, where (u ϵ
F_{0}).
Theorem 9.1: In the lowerlevel set of f, if the elements are x_{1}
≤ x_{2} ≤ x_{3} ≤ …….. ≤ x_{n} for which
f(x_{1}) ≤ f(x_{2}) ≤ f(x_{3}) ≤ ……… ≤ f(x_{n}) ,
then the sequence (f(x_{n}))_{n ϵ N} is convergent and
converges to u ( supremum) .
Proof: By the definition ^{[6]} of the lowerlevel
set of f it is clear that {f(x_{n})} is bounded above by u and since
the sequence (f(x_{n})) is monotonically increasing , so the sequence
is convergent and converges to the supremum u
i.e. its limit is u
.
Similar is the
following theorem.
Theorem 9.2: In the upperlevel set of f, if the elements are x_{1}
≥ x_{2} ≥ x_{3} ≥……..≥ x_{n} for which
f(x_{1}) ≥ f(x_{2}) ≥ f(x_{3}) ≥……… ≥ f(x_{n}) ,
then the sequence (f(x_{n}))_{n ϵ N} is convergent and
converges to u ( infimum ) .
10.
CONCLUSION
Most mathematical
tools for computing, formal modelling and reasoning are crisp,deterministic and
precise in many characters.However several problems in
economics,environment,engineering,social science,medical science etc. do not
always involve crisp data in real life. Consequently,we can not successfully
use the classical method because of various types of uncertainties presented in
the problem.There lies the usefulness of study Fuzzy Mathematics.
The work that has been done through this paper ,I hope, surely create some
impacts on the beginners and motivate them as well.
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A. Kaufmann and M. N. Gupta (1991).  Introduction to Fuzzy Arithmetic Theory and Applications. New York, NY: Van Nostrand Reinhold
G. Alefeld and J. Herzberger (1983). —Introduction to interval computations. New York, NY : Academic Press.
HJ. Zimmermann. (2011) —Fuzzy set theory – and its application. 4th edition. ISBN—978 – 94 – 010 – 0646n – 0.
I. N. Herstein (n.d.).  Topics in Algebra.
John N. Mordeson, Premchand S. Nair (2001).  Fuzzy Mathematics. ISBN – 978 – 3 – 7908 – 1808 – 6. Retrieved from https://link.springer.com/book/10.1007/9783790818086
S. Nanda, N. R. Das (2021). Fuzzy Mathematical concepts
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