Fuzzy Logic  Quick Guide
Fuzzy Logic  Introduction
The word fuzzy refers to things that are unclear or vague. Any event, process or function that changes continuously cannot always be defined as true or false, which means that we have to define such activities in a vague way.
What is fuzzy logic?
Fuzzy logic resembles human decisionmaking methodology. It deals with vague and imprecise information. This is an oversimplification of real world problems and based on degrees of truth rather than the usual true / false or 1/0 logic like Boolean logic.
Take a look at the following diagram. It shows that in fuzzy systems, values are indicated by a number between 0 and 1. Here, 1.0 represents absolute truth and 0.0 represents absolute falsehood . The number that indicatesThis value in fuzzy systems is called the truth value .
In other words, we can say that fuzzy logic is not fuzzy logic, but a logic used to describe the blurry . There can be many other examples like this with the help of which we can understand the concept of fuzzy logic.
Fuzzy logic was introduced in 1965 by Lofti A. Zadeh in his research article “Fuzzy Sets”. He is considered the father of Fuzzy Logic.
Fuzzy Logic  Classical Set Theory
A set is an unordered collection of different elements. It can be written explicitly by listing its elements using the defined parenthesis. If the order of the elements is changed or if an element of a set is repeated, this does not make any change to the set.
Example
 A set of all positive integers.
 A set of all planets in the systemsolar.
 A collection of all the states of India.
 A set of all lowercase letters rs of the alphabet.
Mathematical representation of a set
Sets can be represented in two ways 
List or tabular form
In this form, a set is represented by listing all the elements that compose it. The items are surrounded by braces and separated by commas.
Here are the examples of a set in list or in tabular form 
 Set of vowels in English alphabet, A = {a, e, i, o, u}
 Set of odd numbers less than 10, B = {1,3,5,7,9}
Notation of Set Builder
In this form, the set is defined by specifying a property that the elements of the set have in common. The set is described as A = {x: p (x)}
Example 1  The set {a, e, i, o, u} s 'written
A = {x: x is a vowel in the English alphabet}
Example 2  The set {1,3,5,7,9} is written
B = {x: 1 ≤ x <10 and (x% 2) ≠ 0}
If a element x is a member of any set S, it is denoted x∈S and if an element y is not a member of the set S, it is denoted y∉S.
Example  If S = {1,1.2,1.7,2}, 1 ∈ S but 1,5 ∉ S
Cardinality of a set
The cardinality d A set S, denoted by  S  S , is the number of elements in the set. The number is also called a cardinal number. If a set has an infinite number of elements, its cardinality is ∞∞.
Example   {1,4,3,5}  = 4,  {1,2,3,4,5,…}  = ∞
If there are two sets X and Y,  X  =  Y  denotes two sets X and Y with the same cardinality. This happens when the number of elements in X is exactly equal to the number of elements in Y. In this case, there exists a onetoone function "f" ofX to Y.
 X  ≤  Y  indicates that the cardinality of set X is less than or equal to the cardinality of set Y. This happens when the number of elements in X is less than or equal to that of Y. Here , there is an injective function "f" from X to Y.
 X  < Y  indicates that the cardinality of set X is less than the set The cardinality of Y. This happens when the number of elements in X is less than that of Y. Here, the function " f ”from X to Y is an injective but not a bijective function.
If  X  ≤  Y  and  X  ≤  Y  then  X  =  Y  . The sets X and Y are commonly referred to as equivalent sets .
Types of sets
Sets can be classified into several types; some of which are finite, infinite, subsets, universal, proper, singleton, etc.
Finite set
A set that contains a defined number of elementsis called a finite set.
Example  S = {x  x ∈ N and 70> x> 50}
Infinite set
A set which contains an infinite number of elements is called an infinite set.
Example  S = {x  x ∈ N and x> 10}
Subset
A set X is a subset of set Y (written as X ⊆ Y) if each element of X is an element of the set Y.
Example 1  Let X = {1,2,3,4,5,6} and Y = {1, 2}. Here the set Y is a subset of the set X because all the elements of the set Y are in the set X. Therefore, we can write Y⊆X.
Example 2  Let X = {1,2,3} and Y = {1,2,3}. Here, the set Y is a subset (not a proper subset) of the set X because all the elements of the set Y are in the set X. Therefore, we can write Y⊆X.
Proper subset
The term “proper subset” can be defined as “proper subset”set of but not equal to ”. A set X is a proper subset of the set Y (written as X ⊂ Y) if each element of X is an element of the set Y and  X  <Y.
Example  Let, X = {1,2,3,4,5,6} and Y = {1,2} . Here, set Y ⊂ X, since all the elements of Y are also contained in X and X has at least one element which is more than the set Y.
Universal set
It is a collection of all the elements in a particular context or application. All sets in this context or application are essentially subsets of this universal set. Universal sets are represented by U.
Example  We can define U as the set of all animals on earth. In this case, a set of all mammals is a subset of U, a set of all fish is a subset of U, a set of all insects is a subset of U, and so on.uite.
Empty set or null set
An empty set contains no elements. It is noted Φ. Since the number of elements in an empty set is finite, the empty set is a finite set. The cardinality of an empty set or a null set is zero.
Example  S = {x  x ∈ N and 7
Singleton set or set of units
A singleton set or set of units contains a single element. A set of singleton is denoted {s}.
Example  S = {x  x ∈ N, 7
Equal set
If two sets contain the same elements, they are said to be equal.
Example  If A = {1,2, 6} and B = {6,1,2}, they are equal because each element of set A is an element of set B and each element of set B is an element of set A.
Equivalent set
If the cardinalities of two sets are identical, they are called ensmbles equivalent.
Example  If A = {1,2,6} and B = {16,17,22}, they are equivalent because the cardinality of A is equal to the cardinality by B. ie  A  =  B  = 3
Overlapping set
Two sets which have at least one common element are called overlapping sets. In case of overlapping sets 
$$ n left (A cup B right) = n left (A right) + n left (B right)  n left (A cup B right) $ $
$$ n left (A cup B right) = n left (AB right) + n left (BA right) + n left (A cap B right) $$
$$ n left (Right) = n left (AB right) + n left (A cap B right) $$
$$ n left (B right) = n left (BA right) + n left (A cap B right) $$
Example  Let, A = {1,2,6} and B = {6,12,42}. There is a common element '6 ', so these sets are overlapping sets.
Disjoint set
Two sets A and B are calleds disjoint sets if they don't even have an element in common. Therefore, disjoint sets have the following properties 
$$ n left (A cap B right) = phi $$
$$ n left (A cup B right) = n left (Right) + n left (B rig ht) $$
Example  Let A = {1,2,6} and B = {7,9, 14}, there is not a single common element, therefore these sets are overlapping sets.
Classic set operations
Set operations include Set Union, Set Intersection, Set Difference, Complement of Set and Cartesian Product.
Union
The union of sets A and B (denoted A ∪ BA ∪ B) is the set of elements which are in A, in B, or at the times in A and B. Therefore, A ∪ B = {x  x ∈ A OR x ∈ B}.
Example  If A = {10,11,12,13} and B = {13,14, 15}, then A ∪ B = {10,11,12 , 13,14,15}  The common element appears only once.
Intersection
The intersection ofs sets A and B (denoted by A ∩ B) is the set of elements which are both in A and B. Therefore, A ∩ B = {x  x ∈ A AND x ∈ B}.
Difference / Complement relative
The set difference of sets A and B (denoted A  B) is the set of elements which are only in A but not in B. Therefore, A  B = {x  x ∈ A AND x ∉ B}.
Example  If A = {10, 11,12,13} and B = {13,14,15} , then (A  B) = {10,11,12} and (B  A) = {14,15}. Here, we can see (A  B) ≠ (B  A)
Complement of a set
The complement of a set A (denoted A ′) is the set of elements which are not in the set A. Therefore, A ′ = {x  x ∉ A}.
More precisely, A ′ = (U  A) where U is a universal set which contains all objects.
Example  If A = {x  x belongs to the set of integers add} then A ′ = {y  yn 'does not belong to the set of odd integers}
Cartesian Product / Cross Product
The Cartesian product of n number of sets A1, A2,… An noted A1 × A2 ... × An can be defined as all possible ordered pairs (x1, x2,… xn) where x1 ∈ A1, x2 ∈ A2,… xn ∈ An
Example  If we take two sets A = {a, b} and B = {1,2},
The Cartesian product of A and B is written  A × B = {(a, 1), (a, 2), (b, 1), (b, 2)}
And, the Cartesian product of B and A is written  B × A = {(1, a), (1, b), (2, a), (2, b)}
Properties of classical sets
The properties of sets play a role important role to obtain the solution. Here are the different properties of classic sets 
Commutative property
Having two sets A and B , this property declares 
$$ A cup B = B cup A $$
$$ A cap B = B cap A $$
Associative property
Having three sets A , B and C , this property indicates 
$$ A left cup (B cup C right) = left (A cup B right) cup C $$
$$ A cap left (B cap C right) = left (A cap B right) cap C $$
Distributive property
Having three sets A , B and C , this property indicates 
$$ A cup left (B cap C right) = left (A cup B right) cap left (A cup C right) $$
$$ A cup left (B cup C right) = left (A cup B right) cup left (A cup C right) $$
Idempotence property
For any set A , this property indicates 
$$ A cup A = A $$
$$ A cap A = A $$
Identity property
For set A and set universal X , this property indicates 
$$ A cup varphi = A $$
$$ A cap X = A $$
$$ A cap varphi = varphi $$
$$ A cup X = X $$
Transitive property
Having three sets A , B and C , the property declares 
If $ A subseteq B subseteq C $, then $ A subseteq C $
Involution property
For any set A , this property indicates 
$$ overline {{overline {A}}} = A $$
De Morgan's law
This is a very important law that helps to prove tautologies and contradictions. This law states 
$$ overline {A cap B} = overline {A} cup overline {B} $$
$$ overline {A cup B} = overline {A } cap overline {B} $$
Fuzzy Logic  Set Theory
Fuzzy sets can be seen as an overextension and oversimplification of classical sets. It can be best understood in the context of belonging to a set. Basically, it allows partial membership, which means that it contains items that have varying degrees of membership in the set. From there, we can understand the difference between a classic and a fuzzy ensemble. the set contains componentsents which satisfy precise properties while imprecise fuzziness of belonging.
Mathematical concept
A fuzzy set $ widetilde {A} $ in the universe of the information $ U $ can be defined as a set of ordered pairs and it can be represented mathematically by 
$$ widetilde {A} = left {left (y, mu _ {widetilde {A} } left (y right) right)  y in U right} $$
Here $ mu _ {widetilde {A}} left (y right) $ = degree of membership of $ y $ in widetilde {A}, assume values between 0 and 1, that is, $ mu _ {widetilde {A}} (y) in left [0,1 right] $.
Representation of a fuzzy set
Now consider two cases of information universes and understand how a fuzzy set can be represented.
Case 1
When the universe of information $ U $ is discrete and finite 
$$ widetilde {A} = left {frac {mu _ {widetilde {A}} left (y_1 right )} {y_1} + frac {mu _ {widetilde {A}} left (y_2 right)} {y_2} + frac {mu _ {widetilde {A}} left (y_3 right)} {y_3} + ... right} $$
$ = left {sum_ {i = 1} ^ {n} frac {mu _ {widetilde {A}} left (y_i right)} {y_i} right} $
Case 2
When the universe of information $ U $ is continuous and infinite 
$$ widetilde {A} = left {int frac {mu _ {widetilde {A}} left (y right)} {y} right} $ $
In the representation above, the summation symbol represents the collection of each element.
Operations on fuzzy sets
Having two fuzzy sets $ widetilde {A} $ and $ widetilde {B} $, the universe of information $ U $ and a element ð '¦ of the universe, the following relations express the operation of union, intersection and complement on fuzzy sets.
Union / Fuzzy 'OU '
Consider the following representation to understand how the r workselation Union / Fuzzy 'OU ' 
$$ mu _ {{widetilde {A} cup widetilde {B}}} left (y right) = mu _ { widetilde {A}} vee mu _ widetilde {B} quad forall y in U $$
Here, ∨ represents the "max" operation.
Intersection / Fuzzy "AND"
Consider the following representation to understand how the Intersection / Fuzzy "AND" relationship works 
$$ mu _ {{widetilde {A} cap widetilde {B}}} left (y right) = mu _ { widetilde {A}} wedge mu _ widetilde {B} quad forall y in U $$
Here ∧ represents the "min " operation.
Complement / Fuzzy " NO "
Consider the following representation to understand how the Complement / Fuzzy "NON " 
$$ mu _ {widetilde {A}} relation works = 1  mu _ {widetilde {A}} left (y right) quad y in U $$
Prope rties of fuzzy sets
Let's discuss the different properties of fuzzy sets .
Commutative property
Have two fuzzy sets $ widetilde {A} $ and $widetilde {B} $, this property indicates 
$$ widetilde {A} cup widetilde {B} = widetilde {B} cup widetilde {A} $$
$$ widetilde {A} cap widetilde {B} = widetilde {B} cap widetilde {A} $$
Associative property
Having three fuzzy sets $ widetilde {A} $, $ widetilde { B} $ and $ widetilde {C} $, this property declares 
$$ (widetilde {A} cup left widetilde {B}) cup widetilde {C} right = left widetilde {A} cup ( widetilde {B} right) cup widetilde {C}) $$
$$ (widetilde {A} cap left widetilde {B}) cap widetilde {C} right = left widetilde {A} cup (widetilde {B} right cap widetilde {C}) $ $
Distributive property
Having three fuzzy sets $ widetilde {A} $, $ widetilde {B} $ and $ widetilde {C } $, this property states 
$$ widetilde {A} cup left (widetilde {B} cap widetilde {C} right) = left (widetilde {A} cup widetilde {B} right) cap left (widetilde{A} cup widetilde {C} right) $$
$$ widetilde {A} cap left (widetilde {B} cup widetilde {C} right) = left (widetilde {A} cap widetilde {B } right) cup left (widetilde {A} cap widetilde {C} right) $$
Idempotency property
For any fuzzy set $ widetilde {A} $, this property indicates 
$$ widetilde {A} cup widetilde {A} = widetilde {A} $$
$$ widetilde {A} cap widetilde {A} = widetilde {A } $$
Identity property
For the fuzzy set $ widetilde {A} $ and the universal set $ U $, this property indicates 
$$ widetilde {A} cup varphi = widetilde {A} $$
$$ widetilde {A} cap U = widetilde {A} $$
$ $ widetilde {A} cap varphi = varphi $$
$$ widetilde {A} cup U = U $$
Transitive property
Having three fuzzy sets $ widetilde {A} $, $ widetilde {B} $ and $ widetilde {C} $, this p property states 
$$ If: widetilde {A}subseteq widetilde {B} subseteq widetilde {C},: then: widetilde {A} subseteq widetilde {C} $$
Involution property
For any fuzzy set $ widetilde {A} $, this property indicates 
$$ overline {overline {widetilde {A}}} = widetilde {A} $$
De Morgan's law
This law plays a crucial role in proving tautologies and contradictions. This law states 
$$ overline {{widetilde {A} cap widetilde {B}}} = overline {widetilde {A}} cup overline {widetilde {B}} $$
$$ overline {{widetilde {A} cup widetilde {B}}} = overline {widetilde {A}} cap overline {widetilde {B}} $$
Fuzzy Logic  Function d ' membership
We already know that fuzzy logic is not fuzzy logic but logic that is used to describe fuzziness. This vagueness is best characterized by its membership function. In other words, we can say that the membership function representsnte the degree of truth in fuzzy logic.
Here are some important points about the membership function 

The functions Membership functions were first introduced in 1965 by Lofti A. Zadeh in his first research paper "Fuzzy sets".

Membership functions characterize fuzziness (that is to say all the information of a fuzzy set), whether the elements of fuzzy sets are discrete or continuous.

The functions d Membership can be defined as a technique for solving practical problems through experience rather than knowledge.

Membership functions are represented by graphic shapes.

The rules for defining blur are also blurry.
Mathematical notation
We have already studied that a fuzzy set Ã in the information universe U can be definedini as a set of ordered pairs and it can be represented mathematically by 
$$ wid etilde {A} = left {left (y, mu _ {widetilde {A}} left (y right) right )  y in U right} $$
Here $ mu widetilde {A} left (bullet right) $ = membership function of $ widetilde {A} $; this assumes values between 0 and 1, that is, $ mu widetilde {A} left (bullet right) in left [0,1 right] $. The membership function $ mu widetilde {A} left (bullet right) $ maps $ U $ to the membership space $ M $.
The dot $ left (bullet right) $ in the membership function described above represents the element in a fuzzy set; whether it is discreet or continuous.
Features of membership functions
We will now discuss the different features of membership features.
Core
For any fuzzy set $ widetilde {A} $, the core of a function dBelonging is that region of the universe which is characterized by complete belonging to the whole. Therefore, core consists of all the $ y $ elements of the information universe such as,
$$ mu _ {widetilde {A}} left (y right) = 1 $$
Support
For any fuzzy set $ widetilde {A} $, the support of a membership function is the region of the universe that is characterized by a membership other than zero to the whole. The kernel is therefore made up of all the $ y $ elements of the information universe such as,
$$ mu _ {widetilde {A}} left (y right)> 0 $ $
Frontier
For any fuzzy set $ widetilde {A} $, the limit of a membership function is the region of the universe which is characterized by a membership that is not zero but incomplete on the whole. Therefore the kernel is made up of all $ y $ elements of the information universe such as,
$$ 1> mu _ {widetilde {A}} left (y right)> 0 $$
Fuzzification
It can be defined as the process of transforming a net set into a fuzzy set or a fuzzy set into a set Basically, this operation translates precise and precise input values into linguistic variables.
Here are the two important methods of fuzzification 
Method of fuzzification (sfuzzification )
In this method, the fuzzy set can be expressed using the following relation 
$$ widetilde {A} = mu _1Q left (x_1 right ) + mu _2Q left (x_2 right) + ... + mu _nQ left (x_n right) $$
Here the fuzzy set $ Q left (x_i right) $ is called as the fuzzification kernel . This method is implemented by keeping $ mu _i $ constant and $ x_i $ being transformed into a fuzzy set $ Q left (x_i right) $.
Grade fuzzification (gfuzzification) Method
It is quite similar to the above method but lThe main difference is that it has kept $ x_i $ constant and $ mu _i $ is expressed as a fuzzy set.
Defuzzification
It can be defined as the process of reducing a fuzzy set to a clean set or converting a fuzzy member to a clean member.
We have already studied that the process of fuzzification involves converting net quantities into fuzzy quantities. In a number of engineering applications it is necessary to defuzzify the result or rather "fuzzy result" so that it has to be converted into a net result. Mathematically, the process of Defuzzification is also called "rounding".
The different methods of Defuzzification are described below 
MaxMembership Method
This The method is limited to peak output functions and also known under the name of height method. Mathematically, it can be represented as follows 
$$ mu _ {widetilde {A}} left (x ^ * right)> mu _ {widetilde {A}} left (x right): for: all: x in X $$
Here, $ x ^ * $ is the defuzzified output.
Center of gravity method
This method is also known as the center of surface or center of gravity method. Mathematically, the defuzzified output $ x ^ * $ will be represented by 
$$ x ^ * = frac {int mu _ {widetilde {A}} left (x right) .xdx} {int mu _ {widetilde {A}} left (x right) .dx} $$
Weighted average method
In this method, each membership function is weighted by its value of maximum membership. Mathematically, the defuzzified output $ x ^ * $ will be represented by 
$$ x ^ * = frac {sum mu _ {widetilde {A}} left (overline {x_i} right). overline {x_i}} {sum mu _ {widetilde {A}} left (overline {x_i} right)} $$
Averagemaximum membership
This method is also known as the name of the midmaximy. Mathematically, the defuzzified output $ x ^ * $ will be represented by 
$$ x ^ * = frac {displaystyle sum_ {i = 1} ^ {n} overline {x_i}} {n} $$
Fuzzy Logic  Traditional Fuzzy Refresher
Logic, which was originally just the study of what distinguishes sound argument from unhealthy argument , has now developed into a powerful and rigorous system in which true statements can be discovered, given other statements which are already known to be true.
Predicate logic
This logic deals with predicates, which are propositions containing variables.
A predicate is an expression of one or more variables defined on a specific domain. A predicate with variables can be made a proposition by assigning a value to the variable or by quantifying the variable.
Here are some examples of predicates 
 Let E (x, y) denote "x = y "
 Let X (a, b, c) denote "a + b + c = 0 "
 Let M (x, y) denote "x is married to y "
Propositional logic
A proposition is a set of declarative statements that have a truth value "true " or a "false " truth value. A propositional is made up of propositional variables and connectors. Propositional variables are marked with capital letters (A, B, etc.). Connectors connect the propositional variables.
Some examples of propositions are given below 
 "Man is Mortal ", it returns the value of truth " TRUE "
 " 12 + 9 = 3  2 ", it returns the truth value " FALSE "
The following doesn 't is not a proposition 
Connectives
In propositional logic, we use the following five connectors 
 OR (∨∨ )
 AND (∧∧)
 Negation / NON (¬¬)
 Implication / ifthen (→→)
 If and only if (⇔⇔)
OR (∨∨)
The OR operation of two propositions A and B (written as A∨BA∨B ) is true if at least one of the propositional variables A or B is true.
The truth table is as follows 
A  B  A ∨ B 
True  True  True 
True  False  True 
False  True  True 
False  False  False 
AND ( ∧∧)
The AND operation of two propositions A and B (written as A∧BA∧B) is true if the propositional variables A and B are true.
The truth table is as follows 
A  B  A ∧ B 
True  True  True 
True  False  False 
False  True  False 
False  False  False 
Negation ( ¬¬)
The negation of a proposition A (written as ¬A¬A) is false when A is true and is true when A is false.
The truth table is as follows 
A  ¬A 
True  False 
False  True 
Implication / sialors (→→)
An implication A → BA → B is the proposition “if A, then B”. It is false if A is true and B is false. The other cases are true.
The truth table is as follows 
A  B  A → B 
True  True  True 
True  False  False 
False  True  True 
False  False  True 
If and only if (⇔⇔)
A⇔BA⇔B is a biconditional logical connective which is true when p and q are ditto, i.e. both are false or both two are true.
The truth table is as follows 
A  B  A⇔B 
True  True  True 
True  False  False 
False  True  False 
False  False  True 
Wellformed formula
Wellformed formula (wff) is a predicate containing one of the following 
 All propositional constants and propositional variables are wffs.
 If x is a variable and Y is a wff, ∀xY and ∃xY are also wff.
 The truth value and false values are wffs.
 Each atomic formula is a wff.
 All connectors connecting wffs are wffs.
Quantifiers
The variable in predicates is quantified by quantifiers. There are two types of quantifier in predicate logic 
 Univ quantifierersel
 Existential quantifier
Universal Quantifier
The universal quantifier declares that the s Declarations within its scope are true for each value of the specific variable. It is indicated by the symbol ∀.
∀xP (x) is read as for each value of x, P (x) is true.
Example  "Man is mortal " can be transformed into the propositional form ∀xP (x). Here, P (x) is the predicate which indicates that x is mortal and that the universe of speech is composed only of men.
Existential quantifier
The existential quantifier declares that the statements within its scope are true for certain values of the specific variable. It is indicated by the symbol ∃.
∃xP (x) for some values of x is read as, P (x) is true.
Example  "Some people are dishonest " can be transformed into the propositional form ∃x P (x)where P (x) is the predicate which denotes x is dishonest and the universe of discourse is persons.
Nested quantifiers
If we use a quantifier that appears in the scope of another quantifier, it is called a nested quantifier.
Example
 ∀ a∃bP (x, y) where P (a, b) denotes a + b = 0
 ∀ a∀b∀cP (a, b, c) where P (a, b) denotes a + (b + c) = (a + b) + c
Note  ∀a∃bP (x, y) ≠ ∃a∀bP (x, y)
Fuzzy logic  Approximate reasoning
Here are the different reasoning approximation modes 
Categorical reasoning
In this approximate reasoning mode, the antecedents, not containing fuzzy quantifiers and probabilities fuzzy, are assumed to be in canonical form.
Qualitative reasoning
In this approximate reasoning mode, antecedents and consequents have fuzzy linguistic variables; the rThe inputoutput relationship of a system is expressed as a collection of fuzzy IFTHEN rules. This reasoning is mainly used in the analysis of control systems.
Syllogistic reasoning
In this mode of approximation reasoning, the antecedents of fuzzy quantifiers are linked to the rules of inference. This is expressed as 
x = S _{ 1 } A′s ar Les e B
y = S _{ 2 } C's are D
 
z = S _{ 3 } E's are F
Here, A, B, C, D, E, F are fuzzy predicates.
Dispositional reasoning
In this approximate mode of reasoningtion, the antecedents are arrangements that can contain the fuzzy quantifier "usually". The quantifier Usually relates dispositional and syllogistic reasoning; therefore, it plays an important role.
For example, the rule of projection of inference in dispositional reasoning can be given as follows 
usually (( L, M) is R) ⇒ usually (L is [R ↓ L])
Here [R ↓ L] is the projection of the fuzzy relation R on L
Fuzzy Logic Rule Base
It is a known fact that a human being is always around comfortable making conversations in natural language. The representation of human knowledge can be done using the following natural language expression 
IF antecedent THEN consequent
The expression as shown above is called the Fuzzy rule baseIFTHEN.
Canonical form
Here is the canonical form of Fuzzy Logic Rule Base 
Rule 1  If condition C1, then R1 restriction
Rule 2  If condition C1, then restriction R2
.
.
.
Rule n  If condition C1, then restriction Rn
Interpretations of Fuzzy IFTHEN Rules
Fuzzy rules IFTHEN can be interpreted in the following four forms 
Assignment statements
These types of statements use "=" (equal to the sign) for the purpose of yielding is lying. They are of the following form 
a = hello
climat = summer
Conditional statements
These types of statements use the rule base form" IFTHEN "for the purpose of condition. They are of the formnext 
IF the temperature is high THEN the climate is hot
IF the food is fresh THEN eat.
Unconditional statements
They are of the following form 
GOTO 10
turn the fan off
Linguistic variable
We have studied that fuzzy logic uses linguistic variables which are words or sentences in a natural language. For example, if we say temperature, it is a linguistic variable; whose values are very hot or cold, slightly hot or cold, very hot, slightly hot, etc. The words very, lightly are linguistic hurdles.
Characterization of the linguistic variable
The following four terms characterize the linguistic variable 
 Name of the variable, generally representedby x.
 Set of terms of the variable, generally represented by t (x).
 Syntax rules for generating the values of the variable x.
 Semantic rules to relate each value of x and its meaning.
Fuzzy logic propositions
As we know, propositions are sentences expressed in any language which are usually expressed in the following canonical form 
s like P
Here, s is the subject and P is the predicate.
For example, " Delhi is the capital of India ", this is a proposition where " Delhi " is the subject and " is the capital of India " is the predicate which shows the subject property.
We know that logic is the basis of reasoning and fuzzy logic extends reasoning capacity by using fuzzy predicates, fuzzy predicate modifiers, quantifiersfuzzy and fuzzy qualifiers in fuzzy propositions, which makes the difference with classical logic.
Propositions in fuzzy logic include the following 
Fuzzy predicate
Almost all natural language predicates are fuzzy in nature, therefore, logic fuzzy has predicates like tall, short, hot, hot, fast, etc.
Fuzzy predicate modifiers
We discussed the language hedges above; we also have many fuzzy predicate modifiers which act as covers. They are very essential for producing the values of a linguistic variable. For example, the words very, lightly are modifiers and the sentences might be like " the water is slightly warm . "
Fuzzy quantifiers
This can be defined as a fuzzy number which gives a vague classification of the cardinality of one or more fuzzy or nonfuzzy sets. It can bere used to influence probability in fuzzy logic. For example, words often, most often, are used as fuzzy quantifiers, and clauses can be like " most people are allergic to them ".
Fuzzy qualifiers
Now let's understand fuzzy qualifiers. A fuzzy qualifier is also a proposition of fuzzy logic. The fuzzy qualification has the following forms 
Fuzzy truthbased qualification
It claims the degree of truth of a fuzzy proposition.
Expression  It is expressed as x is t . Here, t is a fuzzy truth value.
Example  (The car is black) is NOT VERY True.
Fuzzy Qualification based on probability
It claims the probability, numerical or interval, of fuzzy proposition.
Expression  It is expressed as x is λ . Here, λ is a probfuzzy ability.
Example  (The car is black) is likely.
Possibilitybased fuzzy qualification
It claims the possibility of a fuzzy proposition.
Expression  It is expressed as x is π . Here, π is a fuzzy possibility.
Example  (The car is black) is almost impossible.
Fuzzy logic  System of inference
Fuzzy inference system is the key unit of a fuzzy logic system with decision making as its main work . It uses the “IF… THEN” rules as well as the “OR” or “AND” connectors to draw the essential decision rules.
Characteristics of the fuzzy inference system
Here are some characteristics of the FIS 

The output of FIS is always a fuzzy set regardless of its input which may be fuzzy or sharp.

It is necessary to have fuzzy output whenit is used as a controller.

A defuzzification unit would be there with FIS to convert fuzzy variables into precise variables.
FIS functional blocks
The following five functional blocks will help you understand the construction of the FIS 

Rule base  Contains fuzzy IFTHEN rules.

Database  It defines the membership functions of fuzzy sets used in fuzzy rules.

Decisionmaking unit  It performs rule operations.

Fuzzification UI Unit  It converts qu antities to fuzzy quantities.

Defuzzification interface unit  It converts fuzzy quantities to precise quantities. Here is a block diagram of the fuzzy interference system.
Operation of the FIS
The operation of the FIS includesthe following steps 

A fuzzification unit supports the application of many fuzzification methods, and converts net input to input blurry.

A knowledge base  the rule base and database collection is formed when converting a clean entry to a fuzzy entry.

The fuzzy defuzzification unit inp ut is finally converted to clean output.
FIS Methods
Now let's talk about the different FIS methods. Here are the two important methods of FIS, having different consequences of fuzzy rules 
 Mamdani 's fuzzy inference system
 Fuzzy model TakagiSugeno (TS method)
Mamdani fuzzy inference system
This system was proposed in 1975 by Ebhasim Mamdani. Basically it was planned to control a combination of steam engine and boiler insynthesizing a set of fuzzy rules obtained from people working on the system.
Steps for calculating the output
The following steps are needed to calculate the output of this FIS 

Step 1  The fuzzy rule set should be determined in this step.

Step 2  In this step, using the input membership function, the input would be blurred.

Step 3  Now establish rule strength by combining fuzzy inputs into fuzzy rules.

Step 4  In this step, determine the consequence of the rule by combining the strength of the rule and the membership function as output .

Step 5  To get the output distribution, combine all the consequents.

Step 6  Finally, a defuzzified output distribution is obtained.
Here is a block diagram of Mamdani Fuzzy System Interface.
Fuzzy TakagiSugeno model (TS method)
This model was proposed by Takagi, Sugeno and Kang in 1985. The format of this rule is given as 
IF x is A and y is B THEN Z = f (x, y)
Here, AB are fuzzy sets in the antecedents and z = f (x, y) is a neat function in the consequent .
Fuzzy inference process
The fuzzy inference process under T The akagiSugeno fuzzy model (TS method) works as follows 

Step 1: Input fuzzification  Here the system inputs are blurred.

Step 2: Apply fuzzy operator  In this step, fuzzy operators need to be applied to get the output.
Rule format of the form Sugeno
The rule format of the form Sugeno is given by 
if 7 = x and 9 = y then the output is z = ax + by + c
Comparison between the two methods
Now let's understand the comparison between the Mamdani system and the Sugeno model.

Output membership function  The main difference between them is in the output membership function . The Sugeno output membership functions are either linear or constant.

Aggregation and defuzzification procedure  The difference between them is also in the consequence of fuzzy rules and due to the same thing, their aggregation and defuzzification procedure also differs.

Mathematical rules  There are more mathematical rules for Sugeno's rule than for Mamdani's rule.

Adjustable parameters  Sugeno controller has more adjustable parameters than M controlleramdani.
Fuzzy Logic  Database and Queries
We have studied in our previous chapters that fuzzy logic is an approach to computation based on " degrees of truth ”rather than on the usual“ true or false ”logic. It deals with rough rather than precise reasoning for solving problems in a way that is more like human logic, hence the process of querying a database through the twovalued realization of l Boolean algebra is not adequate.
Fuzzy scenario of relationships on Databases
The fuzzy scenario of relationships on databases can be understood with the help of the following example 
Example
Suppose we have a database containing records of people who vi located in India. In a simple database we will have the entries made in the following way 
Name  Age  Citizen  Country visited  Days Gone  Year visited 
John Smith  35  US  India  41  1999 
John Smith  35  United States  Italy  72  1999 
John Smith  35  United States  Japan  31  1999 
Now if anyone asks about the person who visited India and Japan in AD 99 and who is a citizen of USA, the output will show two entries with the name John Smith. It is a simple query generating a simple output.
But what if we want to know if the person in the above query is young or not. According to the resultt above, the person 's age is 35 years old. But can we assume that the person is young or not? Likewise, the same can be applied to other fields like past days, year of visit, etc.
The solution to the above problems can be found using fuzzy value sets as follows 

FV (Age) {very young, young, a little old, old}

FV (Days Spent) {just a few days, a few days, a few days, several days}

FV (Year visited) {distant past, recent past, recent}

Now if a query has the fuzzy value, the result will also be fuzzy in nature.
Fuzzy query system
A fuzzy query system is an interface that allows users to get information from the database using sentences in (almost) natural language. Many fuzzy query implementations have beensuggested, resulting in slightly different languages. Although there are some variations depending on the particulars of different implementations, the response to a fuzzy query phrase is usually a list of records, ordered by degree of match.
Fuzzy Logic  Quantification
In the modeling of natural language instructions, quantized statements play an important role. This means that NL is highly dependent on the quantification of the construction which often includes fuzzy concepts like "almost all", "a lot", etc. Here are some examples of proposition quantification 
 Each student has passed the exam.
 Every sports car is expensive.
 Many students passed the exam.
 Many sports cars are expensive.
In the examples above, the quantifiers "All " and "Many " are applied to the restrictions precises "students " as well as "at precise range " (person who has passed the exam "and " cars "as well as " sports "at sharp range.
Fuzzy Events, Fuzzy Averages, and Fuzzy Variations
Using an example, we can understand the above concepts. Suppose we are a shareholder of a company named ABC. And right now, the company is selling each of its shares for ₹ 40. There are three different companies that are similar in business to ABC, but they offer their shares at different rates  ₹ 100 per share, ₹ 85 per share and ₹ 60 per share respectively.
Now the probability distribution of this price takeover is as follows 
Price  100 ₹  85 ₹  60 ₹ 
Probability  0.3  0.5  0.2 
Now fromstandard probability theory, the above distribution averages the expected price as below 
$ 100 × 0.3 + 85 × 0 , 5 + 60 × 0.2 = $ 84.5
And, from standard probability theory, the above gives a variance of the expected price as below 
$ (100  84.5) 2 × 0.3 + (85  84.5) 2 × 0.5 + (60  84.5) 2 × 0.2 = 124.8 $ 25
Suppose that the membership degree of 100 in this set is 0.7, that of 85 is 1, and the degree of membership is 0, 5 for the value 60. This can be reflected in the following fuzzy set 
$$ left {frac {0.7} {100},: frac {1} {85},: frac {0.5 } {60}, right} $$
The fuzzy set obtained in this way is called a fuzzy event.
We want the probability of the fuzzy event for which our calculation gives 
0.7 $ × 0.3 + 1 × 0.5 + 0.5 × 0.2 = 0.21 + 0.5 + 0.1 = $ 0.81
Now we need to calculate the fuzzy mean and the fuzzy variance, the calculation is as follows 
Fuzzy_mean $ = left (frac {1} {0.81} right) × (100 × 0.7 × 0.3 + 85 × 1 × 0.5 + 60 × 0.5 × 0.2) $
$ = $ 85.8
Fuzzy_Variance $ = 7496.91  7361.91 = $ 135.27
Fuzzy logic  Decision making
This is an activity that includes the steps to follow to choose a suitable alternative of these things are necessary to achieve a certain goal.
Stages of decision making
Now let's talk about the stages involved in the decision making process 

Determining the set of alternatives  In this step, the alternatives from which the decision is to be taken should be determined.

Evaluation of alternatives  Here the alternatives must be evaluated so that thedecision can be made on one of the alternatives.

Comparison of alternatives  In this step, a comparison between the evaluated alternatives is performed.
Types of Decision
Making We will now understand the different types of decision making.
Individual decisionmaking
In this type of decisionmaking, only one person is responsible for making the decisions. Such a decisionmaking model can be characterized as 
Objectives and constraints shown above are expressed in terms of fuzzy sets.
Now consider a set A. Then, the goal and the constraints of this set are given by 
$ G_i left (a right) $ = composition $ left [G_i left (a right) right] $ = $ G_i ^ 1 left (G_i left (a right) right) $ with $ G_i ^ 1 $
$ C_j left (a right) $ = composition $ left [C_j left (a right) right] $ = $ C_j ^ 1 left (C_j left (a right) right) $ with $ C_j ^ 1 $ for $ a: in: A $
The fuzzy decision in the above case is given by 
$$ F_D = min [i in X_ {n} ^ {in} fG_i left (a right), j in X_ {m} ^ {in} fC_j left (a right)] $$
Multiperson Decisionmaking
Decisionmaking in this case includes several people so that the expertise m different people are used to make decisions.
The calculation can be given as follows 
Number of people preferring $ x_i $ to $ x_j $ = $ N left (x_i,: x_j right) $
Total number of decision makers = $ n $
Then, $ SC left (x_i,: x_j right) = frac {N left (x_i,: x_j right)} {n} $
Multiobjective decision making
Multigoal decision making occurs when there are multiple goals to achieve. There are the following two problems in this type of decisionmaking 
Mathematically, we can define a universe of n alternatives like 
$ A = left [a_1,: a_2,: ...,: a_i,: ...,: a_n right] $
And the set of "m" goals like $ O = left [o_1,: o_2,: ...,: o_i,: ...,: o_n right] $
Multiattribute decision making
Multiattribute decision making takes place during of the evaluation of alternatives can be carried out according to several attributests of the object. The attributes can be numeric data, linguistic data, and qualitative data.
Mathematically, the multiattribute evaluation is performed based on the linear equation as follows 
$$ Y = A_1X_1 + A_2X_2 + ... + A_iX_i + ... + A_rX_r $$
Fuzzy Logic  Control system
Fuzzy logic is applied with great success in
Why Use Fuzzy Logic in Control Systems
A control system is an arrangement of physical components designed to modify another physical system so that that system has certain desired characteristics. Here are some reasons to useer fuzzy logic in control systems 

In applying traditional control, one must know the objective model and function formulated in precise terms. This makes its application very difficult in many cases.

By applying fuzzy logic for control, we can use human expertise and experience to design a controller.

Fuzzy control rules, essentially IFTHEN rules, can be best used in the design of a controller.
Fuzzy Logic Control (FLC) Assumptions Design
When designing a fuzzy control system, the six basic assumptions following must be done 

The plant is observable and controllable  It must be assumed that the variables of input, output and status are available for observation and control.

Existence of a body of knowledge  It should be assumed that there is a body of knowledge having linguistic rules and a set of inputoutput data from which the rules can be extracted.

Existence of solution  It must be assumed that there is a solution.

A 'good enough ' solution is enough  Control engineering should look for 'a good solution rather than an optimal solution .

Precision range  The fuzzy logic controller must be designed within an acceptable precision range.

Stability and Optimality Issues  Stability and Optimality issues should be open when designing a fuzzy logic controller rather than being addressed explicitly.
Architecture of Fuzzy Logic Control
The following diagram shows the architecture of the Fuzzy Logic Contro1 (FLC).
PriMain components of the FLC
The following are the main components of the FLC, as shown in the figure above 

Fuzzifier  The role of the fuzzifier is to convert sharp input values to fuzzy values.

Fuzzy Knowledge Base  Stores knowledge about all fuzzy inputoutput relationships. It also has the membership function which sets the input variables of the fuzzy rule base and the output variables of the controlled factory.

Fuzzy rule base  It stores knowledge about how the domain process works.

Inference Engine  It acts as a core of any FLC. Basically, it simulates human decisions by doing rough reasoning.

Defuzzifier  The role of the defuzzifier is to convert fuzzy values into unclear valuesheads from fuzzy inference engine.
FLC Design Steps
Here are the steps involved in FLC design 

Identification of variables  Here the input, output and status variables must be identified from the considered installation.

Fuzzy subset configuration  The universe of information is

Getting the membership function  Now get the membership function for each fuzzy subset that we pass to the 'step above.

Configuration of the fuzzy rule base  Now formulate the fuzzy rule base by assigning a relation between the fuzzy input and the fuzzy output.

Fuzzification  The fuzzification process is started at this step.

Combining fuzzy outputs  Applying rough fuzzy reasoning, locate the fuzzy output and merge them.

Defuzzification  Finally, start the defuzzification process to form a
Benefits of fuzzy logic control
Now let's talk about the advantages of fuzzy logic control.

Cheaper  Developing an FLC is comparatively cheaper than developing a modelbased controller or whatever in terms performance.

Rugged  FLCs are more rugged than PID controllers due to their ability to cover a wide range of operating conditions.

Customizable  FLCs are customizable.

Emulate human deductive thinking  FondamenUltimately, FLC is designed to mimic human deductive thinking, the process people use to derive conclusions from what they know.

Reliability  The FLC is more reliable than a conventional control system.

Efficiency  Fuzzy logic offers more efficiency when applied in the control system.
Disadvantages of Fuzzy Logic Control
We will now discuss the disadvantages of Fuzzy Logic Control.

Requires a lot of data  FLC needs a lot of data to be applied.

Useful for moderate historical data  FLC is not useful for programs much smaller or larger than historical data.

Requires great human expertise  This is a disadvantage because the accuracy of the system depends on the experts.these and the expertise of human beings.

Requires regular updating of rules  Rules should be updated over time.
Adaptive fuzzy controller
In this chapter, we will discuss what an adaptive fuzzy controller is and how it works. Adaptive Fuzzy Controller is designed with adjustable parameters as well as a builtin mechanism to adjust them. The adaptive controller has been used to improve the performance of the controller.
Basic steps for implementing the adaptive algorithm
Now let's see the basic steps for implementing the adaptive algorithm.

Collecting observable data  Observable data is collected to calculate controller performance.

Adjusting Controller Parameters  Now with the help of controller performance, the calculationof the controller settings would be done.

Improvement in controller performance  In this step, the controller parameters are adjusted to improve the performance of the controller.
Operational Concepts
Design of a controller is based on an assumed mathematical model that looks like a real system. The error between the real system and its mathematical representation is calculated and if it is relatively insignificant, the model is supposed to work efficiently.
A threshold constant which sets a limit for the efficiency of a controller also exists. Command input is fed into both the real system and the mathematical model. Here, suppose that $ x left (t right) $ is the output of the system rea and $ y left (t right) $ is the output of the mathematical model. Then the error $ epsilon left (t right) $ can be calculated as follows 
$$ epsilon left (t right) = x left (t right)  y left (t right) $$
Here, the desired $ x $ is the output we want from the system and $ mu left (t right) $ is the output from the controller going to both the real and math model.
The following diagram shows how the error function is tracked between the output of a real system and the mathematical model 
System settings
A fuzzy controller whose the design is based on the fuzzy mathematical model will have the following form of fuzzy rules 
Rule 1  IF $ x_1 left (t_n right) in X_ {11}: AND. .. AND: x_i left (t_n right) in X_ {1i} $
THEN $ mu _1 left (t_n right) = K_ {11 } x_1 left (t_n right) + K_ {12} x_2 left (t_n right): + ... +: K_ {1i} x_i left (t_n right) $
Rule 2  IF $ x_1 left (t_n right) in X_ {21}: AND ... AND: x_i left (t_n right) in X_ {2i} $
THEN $ mu _2 left (t_n right) = K_ {21} x_1 left (t_n right) + K_ {22} x_2 left (t_n right): + ... +: K_ {2i} x_i left (t_n right) $
.
.
.
Rule j  IF $ x_1 left (t_n right) in X_ {k1}: AND ... AND: x_i left (t_n right) in X_ {ki} $
THEN $ mu _j left (t_n right) = K_ {j1} x_1 left (t_n right) + K_ {j2} x_2 left (t_n right): + ... +: K_ {ji} x_i left (t_n right) $
The above set of parameters characterizes the controller.
Mechanism tuning
Controller parameters are adjusted ted to improve controller performance. The process of calculating the adjustment of parameters is the adjustment mechanism.
Mathematically, let $ theta ^ left (n right) $ be a set of parameters to be adjusted at time $ t = t_n $.The adjustment can be the recalculation of parameters,
$$ theta ^ left (n right) = Theta left (D_0,: D_1,: ...,: D_n right) $$
Here $ D_n $ are the data collected at time $ t = t_n $.
Now this formulation is reformulated by updating the parameter set to its previous value as,
$$ theta ^ left (n right) = phi (theta ^ {n1},: D_n) $$
Parameters for selecting an adaptive fuzzy controller
The following parameters must be taken into account to select an adaptive fuzzy controller 

Can the system be approached entirely by a fuzzy model?

If a system can be approached entirely by a fuzzy model, are the parameters of that fuzzy model readily available or do they have to be determined online?

If a system cannot be fully approximated by a fuzzy model, can it be approximated by monetworks by a set of fuzzy model?

If a system can be approached by a set of fuzzy models, do these models have the same format with different parameters or do they have different formats?

If a system can be approached by a set of fuzzy models having the same format, each with a different set of parameters, are these sets of parameters readily available or should they be can they be determined online?
Blur in Neural Networks
The Artificial Neural Network (ANN) is a network of efficient computing systems from which the central theme is borrowed to the analogy of biological neural networks. ANNs are also referred to as "artificial neural systems", "parallel distributed processing systems", "connectionist systems". ANN acquires a large collection of units which are interconnected in some patt ern to allow communications between units.These units, also called nodes or neurons, are simple processors that work in parallel.
Each neuron is connected to another neuron via a connection link. Each connection link is associated with a weight containing information on the input signal. This is the most useful information for neurons to solve a particular problem because the weight usually inhibits the signal that is being communicated. Each neuron has its internal state which is called the activation signal. The output signals, which are produced after combining the input signals and the activation rule, can be sent to other units. It also consists of a 'b ' bias whose weight is always 1.
Why use fuzzy logic in a neural network
As we saw above, each neuron in ANN is connected to another neuron via a connecting link and this link is associated with a weight containing information about the d 'Entrance. Therefore, we can say that the weights have some useful information about the inputs to solve the problems.
Here are some reasons to use fuzzy logic in neural networks 

Fuzzy logic is widely used for define weights, from fuzzy sets, in neural networks.

When precise values are not possible to apply, then fuzzy values are used.

We have already studied that training and learning help neural networks to function better in unexpected situations. At that time, fuzzy values would be more applicable than net values.

When we use fuzzy logic in neural networks, the values don't have to be sharp and the processing can be done in parallel.
Fuzzy Cognitive Map
It 'sa form of blur ins neural networks. Basically, FCM is like a dynamic state machine with fuzzy states (not just 1 or 0).
Difficulty in using fuzzy logic in neural networks
Despite many advantages, there are also some difficulties when using fuzzy logic in neural networks. The difficulty is linked to the membership rules, to the need to build a fuzzy system, because it is sometimes complicated to deduce it with the given set of complex data.
Neural fuzzy logic
The inverse relationship between neural network and fuzzy logic, i.e. the neural network used to train fuzzy logic, is also a good area of study. Here are two major reasons for building neuraltrained fuzzy logic 

New data models can be easily learned using neural networks, by therefore it canbe used to preprocess data in fuzzy systems.

The neural network, due to its ability to learn new relationships with new input data, can be used to refine fuzzy rules to create a system adaptive blur.
Examples of Fuzzy NeuralTrained System
Fuzzy NeuralTrained Systems are used in many commercial applications. Now let's see some application examples of the Fuzzy NeuralTrained System 

The Laboratory for International Fuzzy Engineering Research (LIFE) in Yokohama, Japan owns a backpropagating neural network that derives from fuzzy rules. This system has been successfully applied to the currency trading system with about 5000 fuzzy rules.

Ford Motor Company has developed fuzzy drivable systems for the idle control of automobiles.

NeuFuz, a software product of National Semiconductor Corporation, supports fuzzy rule generation with a neural network for control applications.

AEG Corporation of Germany uses neural driven fuzzy control system for its water and energy saving machine. It has a total of 157 fuzzy rules.
Fuzzy Logic  Applications
In this chapter, we will discuss areas where the concepts of Fuzzy Logic are widely applied.
Aerospace
In aerospace, fuzzy logic is used in the following areas 
 Altitude control spacecraft
 Satellite altitude control
 Flow and mixture regulation in aircraft deicing vehicles
Automobile
In automobiles, fuzzy logic is used in the following areas 
 Drivable fuzzy systems pfor idle control
 Method of programming gear changes for automatic transmission
 Intelligent highway systems
 Traffic control
 Improving the efficiency of automatic transmissions
Business
In business, fuzzy logic is used in the following areas 
 Decision support systems
 Staff assessment in a large company
Defense
In defense, fuzzy logic is used in the following areas 
 Underwater target recognition
 Automatic target recognition of thermal infrared images
 Naval decision support
 Control of a hypervelocity interceptor
 Fuzzy overall modeling of NATO decision making
Electronics
In electronics, fuzzy logic is used in dfollowing areas 
 Automatic exposure control in video cameras
 Humidity in a clean room
 Air conditioning systems
 Programming the washing machine
 Microwave ovens
 Vacuum cleaners
Finance
In finance, fuzzy logic is used in the following areas 
 Banknote transfer control
 Fund management
 Stock market forecast
Industrial sector
In industry, fuzzy logic is used in the following areas 
Control of cement kiln Control of heat exchanger Control of wastewater treatment process of activated sludge Control of sewage treatment plant Quantitative analysis of models for industrial quality assurance Control of satisfaction problems ofs constraints in structural design Control of water treatment plants Manufacturing
In the manufacturing industry, the fuzzy logic is used in the following areas 
 Optimization of cheese production
 Optimization of milk production
Marine
In the maritime domain, fuzzy logic is used in the following areas 
 Autopilot for ships
 Optimal route selection
 Control of autonomous underwater vehicles
 Pilotage of ships
Medical
In the medical field, fuzzy logic is used in the following fields 
 Medical diagnostic aid system
 Control of blood pressure during anesthesia
 Multivariate control of anesthesia
 Modeling of neuropathological resultses in patients with Alzheimer 's disease
 Radiological diagnostics
 Diagnosis by fuzzy inference of diabetes and prostate cancer
Securities
In securities, fuzzy logic is used in the following areas 
 Decision systems for securities trading
 Various security devices
Transport
In transport, fuzzy logic is used in the following areas 
 Automatic metro operation
 Control of train tibles
 Rail acceleration
 Braking and stopping
Pattern recognition and classification
In Pattern Recognition and classification, fuzzy logic is used in the following areas 
 Speech recognition based on fuzzy logic
 Based on fuzzy logic
 Speech recognitionhandwriting
 Analysis of facial features based on fuzzy logic
 Analysis of commands
 Search for fuzzy images
Psychology
In psychology, fuzzy logic is used in the following fields 
 Analysis of human behavior based on fuzzy logic
 Criminal investigation and prevention based on fuzzy logic reasoning