Suppose f : X → [ 0 , 1 ] is a fuzzy set, then the range of f is defined as f ( x ) = { f ( x ) : x ∈ X } [2].

Example: Suppose X = { 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 } and f ( x ) = 1 3 x is a fuzzy set on X , then the range of f is given in the following table:

Suppose A : X → [ 0 , 1 ] is a fuzzy set, then the height of A is denoted by h ( A ) and is defined as h ( A ) = ∨ A ( x ) = Sup { μ A ( x ) : x ∈ X } . This is the highest membership grade obtained by any element in that set [2].

Suppose A : X → [ 0 , 1 ] is a fuzzy set. Then the “Support of A ” is the ordinary subset of X , that is the support of fuzzy set A within a universal set X is the crisp set that contains all the elements of X that have nonzero membership grades in A . Symbolically, supp   A = { x ∈ X : μ A ( x ) > 0 } [2][3].

Suppose A : X → [ 0 , 1 ] is a fuzzy set. Then “Core of A ” is denoted by Core ( A ) and defined as follows: Core ( A ) = { x ∈ X : μ A ( x ) = 1 } [2]. Core is shown into Figure 2.

Figure 2. Core of fuzzy set.

Suppose A : X → [ 0 , 1 ] is a fuzzy set. Then A is called a normal fuzzy set if and only if h ( A ) = ∨ A ( x ) = Sup { μ A ( x ) } = 1 [2].

Example: Suppose X = [ 3 , 50 ] and A : X → [ 0 , 1 ] is a fuzzy set, where μ A ( x ) = 3 x . Then μ A ( x ) = [ 0.06 , 1 ] .Therefore h ( A ) = 1 , hence A is a normal fuzzy set.

Suppose B : X → [ 0 , 1 ] is a fuzzy set. Then B is called a sub normal fuzzy set if an only if h ( B ) = ∨ B ( x ) = Sup { μ B ( x ) } < 1 [2]. Normal and Subnormal fuzzy set is shown into Figure 3.

Figure 3. Normal and subnormal fuzzy set.

Example: Suppose X = [ 3 , 50 ] and B is a fuzzy set where μ B ( x ) = x x + 3 . Then μ B ( x ) = [ 0.5 , 0.94 ] . Therefore h ( B ) = 0.94 , hence B is a sub normal fuzzy set.

One of the most important concepts of fuzzy sets is the concept of α -cut and a strong α -cut. Given a fuzzy set A defined on X and any number α ∈ [ 0 , 1 ] , the α -cut, A α , and the strong α -cut, A α + , are the crisp sets A α = { x : A ( x ) ≥ α } and A α + = { x : A ( x ) > α } [2]. That is, the α -cut (or the strong α -cut) of a fuzzy set A is the crisp set A α (or the crisp set A α + ) that contains all the elements of the universal set X whose membership grades in A are greater than or equal to (or only greater than) the specified value of α . It is shown into Figure 4.

Figure 4.α-cut of fuzzy set.

Example: Suppose M : X → [ 0 , 1 ] defines a fuzzy set of “Middle aged people” where, X is the set of ages of human beings, such that

Let α = 0.5 then M 0.5 = [ 27.5 , 52.5 ] and M 0.5 + = ( 27.5 , 52.5 ) .

The standard complement A ¯ , of fuzzy set A with respect to the universal set X is defined for all x ∈ X by the equation A ¯ ( x ) = 1 − A ( x ) [2].

Example: Suppose X = [ 0 , 3 ] and A : X → [ 0 , 1 ] is a fuzzy set, where A ( x ) = 1 1 + 10 x . Then the membership function of complement of A is given as A ¯ ( x ) = 1 − 1 1 + 10 x = 10 x 1 + 10 x . Complement of fuzzy set is shown into Figure 5.

Figure 5. Complement of fuzzy set.

Given two fuzzy sets A and B defined on the universal set X . Then the “standard intersection” of A and B is denoted by A ∩ B and defined by ( A ∩ B ) ( x ) = min { A ( x ) , B ( x ) } ; ∀ x ∈ X [2].

On the other hand, the “standard union” of two fuzzy sets A and B is denoted by A ∪ B and defined by ( A ∪ B ) ( x ) = max { A ( x ) , B ( x ) } ; ∀ x ∈ X [2].

Example: Consider the fuzzy sets A and B defined on the interval X = [ 0 , 10 ] of real numbers by the membership grade functions A ( x ) = x x + 2 and B ( x ) = 2 − x . The graphs of the membership grade functions of ( A ∩ B ) ( x ) and ( A ∪ B ) ( x ) is shown in Figure 6.

Suppose A : X → [ 0 , 1 ] and B : X → [ 0 , 1 ] are two fuzzy sets, then we say that B is a fuzzy subset of A and write B ⊆ A , if and only if B ( x ) ≤ A ( x ) ; ∀ x ∈ X [2].

(a) (b)

Figure 6. Intersection and Union of fuzzy.

Example: Suppose X = { 1 , 2 , 3 , 4 , 5 } and A : X → [ 0 , 1 ] , B : X → [ 0 , 1 ] are two fuzzy sets on X , where A ( x ) = 1 x and B ( x ) = 1 x 2 , then

Clearly, ∀ x ∈ X ; B ( x ) ≤ A ( x ) Therefore B ⊂ A .

A fuzzy set A is convex if and only if it’s α -cuts are convex [2][3]. An equivalent definition of convexity is that the fuzzy set A on ℝ is convex if and only if ∀ x 1 , x 2 ∈ X and ∀ λ ∈ [ 0 , 1 ] , μ A ( λ x 1 + ( 1 − λ ) x 2 ) ≥ min [ μ A ( x 1 ) , μ A ( x 2 ) ] [3].

Example: Suppose X = [ 0 , 5 ] and μ A : X → [ 0 , 1 ] is a fuzzy set, where μ A ( x ) = 1 1 + 10 ( x − 2 ) 2 . Then it is a convex fuzzy set shown in Figure 7.

A function t : I × I → I is called t-norm if it satisfies the following properties:

[ t 1 ] : a ≤ b , c ≤ d ⇒ t ( a , c ) ≤ t ( b , d ) [Monotonicity]

[ t 2 ] : t ( a , b ) = t ( b , a ) [Commutativity]

[ t 3 ] : t ( a , t ( b , c ) ) = t ( t ( a , b ) , c ) [Associativity]

[ t 4 ] : t ( a , 1 ) = a [Boundary condition]

∀ a , b , c , d ∈ I ; Where I = [ 0 , 1 ] and set of all t-norms will be dented by t N . All the t-norms are presented in Figure 8.

3.1.1. Algebraic Product

The t-norm Algebraic product will be denoted by t a and defined by t a ( x , y ) = x y ; ∀ ( x , y ) ∈ I × I .

3.1.2. Bounded Product

The t-norm Bounded product will be denoted by t b and defined by ∀ ( x , y ) ∈ I × I ;

(a) (b)

(c) (d)

Figure 8. Fuzzy intersections: t-Norms.

3.1.3. Drastic Product

The t-norm drastic intersection or drastic product will be denoted by t d and define by ∀ ( x , y ) ∈ I × I ; t d ( x , y ) = { x ; y = 1 y ; x = 1 0 ; otherwise .

3.1.4. Standard Intersection

The t-norm standard intersection or Min will be denoted by t m and defined by ∀ ( x , y ) ∈ I × I ; t m ( x , y ) = min ( x , y ) = x ∧ y = { x ; x ≤ y y ; y ≤ x .

A function u : I × I → I is called t-conorm if it satisfies the following properties:

[ u 1 ] : a ≤ b , c ≤ d ⇒ u ( a , c ) ≤ u ( b , d ) [Monotonicity]

[ u 2 ] : u ( a , b ) = u ( b , a ) [Commutativity]

[ u 3 ] : u ( a , u ( b , c ) ) = u ( u ( a , b ) , c ) [Associativity]

[ u 4 ] : u ( a , 0 ) = a [Boundary condition]

∀ a , b , c , d ∈ I ; where I = [ 0 , 1 ] . All the t-conorms are presented in Figure 9.

3.2.1. Algebraic Sum

The t-conorm Algebraic sum will be denoted by u a and defined by u a ( x , y ) = x + y − x y ; ∀ ( x , y ) ∈ I × I .

3.2.2. Bounded Sum

The t-conorm Bounded sum will be denoted by u b and defined by ∀ ( x , y ) ∈ I × I ; u b ( x , y ) = min ( 1 , x + y ) = 1 ∧ ( x + y ) = { x + y ; x + y ≤ 1 1 ; x + y ≥ 1 .

(a) (b)

(c) (d)

Figure 9. Fuzzy unions: t-conorms.

3.2.3. Drastic Union

The t-conorm drastic union or drastic sum will be denoted by u d and define by ∀ ( x , y ) ∈ I × I ; u d ( x , y ) = { x ; y = 0 y ; x = 0 1 ; otherwise .

3.2.4. Standard Union

The t-conorm standard union or Max will be denoted by u m and defined by ∀ ( x , y ) ∈ I × I ; u m ( x , y ) = max ( x , y ) = x ∨ y = { x ; x ≥ y y ; x ≤ y

Among the various shapes of fuzzy number, triangular fuzzy number (TFN) is the most popular one [9]. It is a fuzzy number represented with three points A = ( a 1 , a 2 , a 3 ) where the fuzzy membership function (fmf) of A is given by,

The α -cut is written as A α = [ ( a 2 − a 1 ) α + a 1 , − ( a 3 − a 2 ) α + a 3 ] ; ∀ α ∈ [ 0 , 1 ] for Triangular fuzzy number. Triangular fuzzy number is shown in Figure 10.

Figure 10. Triangular fuzzy number.

Another shape of fuzzy number is trapezoidal fuzzy number [9]. This shape is originated from the fact that there are four points whose membership degree is maximum ( α = 1 ) . A = ( a 1 , a 2 , a 3 , a 4 ) is called a trapezoidal fuzzy number with the fuzzy membership function (fmf) given as follows:

The α -cut is written as A α = [ ( a 2 − a 1 ) α + a 1 , − ( a 4 − a 3 ) α + a 4 ] ; ∀ α ∈ [ 0 , 1 ] for Trapezoidal fuzzy number. Trapezoidal fuzzy number is shown in Figure 11.

Fuzzy arithmetic is based on two properties of fuzzy numbers: (a) each fuzzy set, and thus also each fuzzy number, can fully and uniquely be represented by it’s α -cut; and (b) α -cut of each fuzzy number are closed intervals of real numbers for all α ∈ ( 0 , 1 ] . These properties enable us to define arithmetic operations on fuzzy numbers in terms of arithmetic operations on their α -cut [2]. Let “ ∗ ” denote any of the four arithmetic operations on closed intervals: addition (+), subtraction (−), multiplication (∙) and division (/). Then,

[ a , b ] ∗ [ d , e ] = { f ∗ g : a ≤ f ≤ b , d ≤ g ≤ e } is a general property of all arithmetic operations on closed intervals [2]. The four arithmetic operations on closed intervals are defined as follows:

1) Addition: [ a , b ] + [ d , e ] = [ a + d , b + e ]

2) Subtraction: [ a , b ] − [ d , e ] = [ a − e , b − d ]

3) Multiplication: [ a , b ] ⋅ [ d , e ] = [ min ( a d , a e , b d , b e ) , max ( a d , a e , b d , b e ) ]

4) Division [ a , b ] / [ d , e ] = [ a , b ] ⋅ [ 1 e , 1 d ] = [ min ( a / d , a / e , b / d , b / e ) , max ( a / d , a / e , b / d , b / e ) ]

In the crisp equation a x + b = c , we obtain the solution x = ( c − b ) a , if a ≠ 0 considering the facts ( b − b ) = 0 and ( 1 a ) a = 1 . But if we are trying to apply the same approach with the fuzzy equation A ¯ ⋅ X ¯ + B ¯ = C ¯ , we get ( 1 A ¯ ) ( A ¯ X ¯ + ( B ¯ − B ¯ ) ) = ( 1 A ¯ ) ( C ¯ − B ¯ ) , from where the solution is not possible since ( B ¯ − B ¯ ) ≠ 0 and ( 1 A ¯ ) ( A ¯ ) ≠ 1 .

One way to solve the fuzzy linear equation A ¯ ⋅ X ¯ + B ¯ = C ¯ is to simply fuzzify the crisp solution ( c − b ) a , a ≠ 0 . The fuzzified crisp solution is C ¯ − B ¯ A ¯ , where assuming that zero does not belong to the support of A ¯ .

If X ¯ e is the value of C ¯ − B ¯ A ¯ then by using extension principle [2], we get X ¯ e = max { π ( a , b , c ) | ( c − b ) a = x } where π ( a , b , c ) = min { A ( a ) , B ( b ) , C ( c ) }

Since the expression c − b a , a ≠ 0 , is continuous in a , b , c we know how to find α – cut of X ¯ e .

“DARAZ”¹ bought 10,000 thermal scanners at tk. 5000 each. He wishes to sell around 40% of them for a profit of approximately 25%. The problem is that at what price should the dealer sell the rest if he wants his average profit for all the items to be about 40%?

Solution. If sell percent, a = 40 % = 0.40 ; percentages of profit, b = 25 % = 0.25 ; percentages of average profit, c = 40 % = 0.40 and setting price of rest of the item = p , then the crisp equation of this problem be like ( 10000 a ) ( 5000 ) b + ( 10000 − 10000 a ) ( p − 150 ) = ( 10000 ) 5000 c . ⇒ p = 5000 ( 1 + c − a b 1 − a ) = 7500 .

That means the dealer should set a fixed selling price tk. 7500 for each of the rest of the items. But in reality it is very difficult to determine such a specific price for rest of the products. Now we construct the fuzzy equation of the given problem by considering three triangular fuzzy numbers A ¯ = ( 0.30 / 0.40 / 0.50 ) , B ¯ = ( 0.15 / 0.25 / 0.35 ) and C ¯ = ( 0.35 / 0.40 / 0.45 ) . Where A ¯ , B ¯ , C ¯ represent the sell percentage, percentages of profit and percentages of average profit respectively. Hence the fuzzy equation to solve is

Here P ¯ is the expected sell price for the rest of products. We look forward to the solution P ¯ by using the method discussed above.

Let A ¯ α = [ 0.30 + 0.10 α , 0.50 − 0.10 α ] , B ¯ α = [ 0.15 + 0.10 α , 0.35 − 0.10 α ] and C ¯ α = [ 0.35 + 0.05 α , 0.45 − 0.05 α ] where α ∈ [ 0 , 1 ] ; also assume that P ¯ > 5000 . The α -cuts of P ¯ are:

Now defuzzify this solution by putting the values of α = 0 and α = 1 we get, P ¯ = ( 6750 / 7500 / 8750 ) is also a triangular fuzzy number. Membership function of the solution is given as follows:

Graphical presentation of this solution is given in Figure 12.

Bangladesh has a warm and humid climate. It experiences heavy rain and tropical cyclones. The average temperature is around 26˚C, but it can range from 15˚C to 34˚C during the year [14]. The warmest months are from April to September, which is also the rainy season. The winter months, from December to February, are cooler and drier. Bangladesh gets about 2200 millimeters (mm) of rain each year. Most areas receive at least 1500 mm, while some northeastern regions get up to 5000 mm of rain annually. Humidity is high all year, especially during the monsoon season from June to October. The Southwest monsoon brings warm and moist air from the Indian Ocean, causing heavy rain. On average, a tropical cyclone classified as a Tropical Storm or stronger hits Bangladesh every two to three years. These storms bring heavy rain, strong winds, and storm surges [14].

Figure 12. Membership function of profit.

Rainfall Prediction

Weather systems can be influenced by many factors, which can be difficult to model precisely. Fuzzy logic can help categorize conditions into fuzzy sets like “light rain,” “moderate rain,” and “heavy rain.” When certain meteorological indicators (like temperature, humidity, and atmospheric pressure) fall within these fuzzy ranges, forecasts can be generated that express the likelihood of different rainfall levels, accommodating uncertainty. Forecasting rainfall using fuzzy logic involves defining a fuzzy membership function to represent the degree of membership of rainfall data in specific linguistic categories (e.g., “low,” “moderate,” “high”). We construct a fuzzy membership function for rainfall forecasting, let the rainfall be classified into three categories:

1) Low rainfall (<0.5 mm per hour)

2) Moderate rainfall (<2 mm per hour)

3) High rainfall (>4 mm per hour)

Assume rainfall values range between 0 mm (no rainfall) and 4 mm (very heavy rainfall) per hour. We define Fuzzy Membership Functions are as follows:

1) Low Rainfall ( A 1 ): μ A 1 ( x ) = { 1 ; 0 ≤ x ≤ 0.5 2 − x 1.5 ; 0.5 < x ≤ 2 0 ; x > 2

2) Moderate Rainfall ( A 2 ): μ A 2 ( x ) = { 0 ; x < 0.5 or x > 4 x − 0.5 1.5 ; 0.5 ≤ x ≤ 2 4 − x 2 ; 2 < x ≤ 4

3) High Rainfall ( A 3 ): μ A 3 ( x ) = { 0 ; x ≤ 2 x − 2 2 ; 2 < x < 4 1 ; x ≥ 4

Classification of rainfall is shown in Figure 13.

Figure 13. Degree of rainfall.

Fuzzy set theory allows us to handle imprecise information, making it ideal for medical diagnoses such as diabetes classification. Instead of categorizing patients strictly as “diabetic” or “non-diabetic,” fuzzy sets allow for gradual transitions based on membership functions.

Fuzzy Set Representation of Diabetes

Let B be the set of blood sugar levels (in mg/dL). We define a fuzzy set “Diabetic” ( D ) with a membership function μ D ( b ) that assigns a degree of membership to each blood sugar level. We classify blood sugar levels into three fuzzy sets:

1) Normal Blood Sugar (Non-Diabetic)

2) Pre-diabetic (Borderline Case)

3) Diabetic (High Risk)

Now fuzzy membership function for Diabetes can be defined as follows:

1) Normal Blood Sugar: μ D ( b ) = 1 for b ≤ 100 , gradually decreasing to 0 for 100 ≤ b < 125 .

2) Pre-diabetic: μ D ( b ) = 0 for b ≤ 100 , gradually increasing to 1 at b = 125 , then decreasing to 0 at b = 140 .

3) Diabetic: μ D ( b ) = 0 for b ≤ 125 , gradually increasing to 1 for b ≥ 140 .

Mathematical Representation:

Graphical representation of this classification is shown in Figure 14.

Figure 14. Diabetic classification according to degree of membership.

Sample Calculation: If a patient has a fasting blood sugar level of 130 mg/dL, then

Normal: μ N ( 130 ) = 0 (not in the normal range)Pre-diabetic: μ P ( 130 ) = 140 − 130 15 = 0.67 Diabetic: μ D ( 130 ) = 130 − 125 15 = 0.33

Thus, the patient has a 67% membership in the pre-diabetic category and a 33% membership in the diabetic category.