How Fuzzy Logic Solves Uncertain Problems in Maths
The term fuzzy refers to things that are not clear or are in vague form. In the real world, most of the times we encounter a situation where we can’t determine whether the given state is true or false, then fuzzy logic provides valuable flexibility for reasoning. In this way, we can consider the incorrectness and uncertainties of any situation. So it is an approach for variable processing that allows for multiple values to be processed through the same variable. Sometimes there is a situation in real life, where we cannot decide that the given problem or statement is either true or false. At that point in time, the concept of fuzzy provides many values between true and false and gives the flexibility to find the best solution to that problem.
In this article, we will learn the concept of fuzzy logic, fuzzy sets and fuzzy logic and different types of fuzzy sets along with algorithm and architecture.
History of Fuzzy Logic Systems
Although, the concept of fuzzy logic had been introduced in the 1920s. The term fuzzy logic was first used in 1965 by Lotfi Zadeh, a professor at UC Berkeley in California. He observed that conventional computer logic was not capable of manipulating data that are represented by subjective or unclear human ideas.
Fuzzy Logic Meaning
Fuzzy logic comes from mathematics that helps us to study concepts of fuzzy, which also involves fuzzy sets of data. Mathematicians may use different types of terms when they are referring to fuzzy concepts and fuzzy analysis. Broadly these terms can be classified as fuzzy semantics. Fuzzy logic meaning is to attempt and solve problems with an open, imprecise spectrum of data which makes it possible to obtain an array of accurate conclusions. Fuzzy logic is designed to solve problems by considering all available information and makes the best possible decision from the given input.
Fuzzy Logic Definition
In the boolean system truth value, 1.0 represents absolute truth-value whereas 0.0 represents the absolute false value. If we consider the fuzzy system, there is no logic for the absolute truth and absolute false value. If we refer to fuzzy logic, there is an intermediate value to present which is partially true and partially false.
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In practice, these constructs allow for multiple values of the "true" condition. Instead of True, the numerical equivalent is 1 and False whose equivalent is 0 (or vice versa). The true condition could be any number having values less than one and greater than zero. With the help of these algorithms to make decisions based on ranges of price data as opposed to one discrete data point.
Why Do We Need Fuzzy Logic?
Fuzzy logic is mostly used for commercial and practical purposes.
It can control machines and consumer products.
It may not give accurate reasoning, but it can provide acceptable reasoning.
Fuzzy logic helps to deal with different types of uncertainty in engineering.
Fuzzy Logic System
Fuzzy Logic Systems Architecture
It has four main parts as shown :
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Fuzzification Module − It transforms the system inputs, which are in crisp numbers, into fuzzy sets. We can split the input signal into five steps such as −
LP - x is Large Positive
MP - x is Medium Positive
S - x is Small
MN - x is Medium Negative
LN - x is a Large Negative
Knowledge Base − It stores IF-THEN rules that are provided by the experts.
Inference Engine − It simulates the human reasoning process by making fuzzy inference on the inputs and IF-THEN rules.
De fuzzification Module − It transforms the fuzzy set that is obtained by the inference engine into a crisp value.
Example of a Fuzzy Logic System
Let us consider an air conditioning system that has a 5-level of a fuzzy logic system. This system can adjust the temperature of the air conditioner by comparing the room temperature and the target temperature value.
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Advantages of Fuzzy Logic System
This system is applicable for any type of inputs whether it is imprecise, distorted or noisy input information.
The construction of Fuzzy Logic Systems is easy and in an understandable form.
Fuzzy logic is a mathematical concept of set theory and the reasoning of that is quite simple.
It provides a very efficient solution to complex problems in all fields of life because it resembles human reasoning and decision making.
The algorithms can be described with the help of little data, so little memory is required.
The structure of Fuzzy Logic Systems is designed in such a way that it is easy and understandable.
Fuzzy logic in AI helps you to control machines and consumer products.
It may not offer accurate reasoning, but the only acceptable reasoning.
Fuzzy logic in Data Mining helps you to deal with the uncertainty in engineering.
Mostly robust as no precise inputs required.
It can be programmed in the situation when the feedback sensor stops working.
It can easily be modified to improve or alter the system performance.
inexpensive sensors can be used which helps you to keep the overall system cost and complexity low.
It provides the most effective solution to complex issues.
The Architecture of Fuzzy Logic
Its Architecture contains four parts :
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Rule Base: It contains the set of rules and the IF-THEN conditions provided by the experts to govern the decision-making system, based on linguistic information. Recent developments in fuzzy theory provide several effective methods for the design and tuning of fuzzy controllers. Most of these developments help us to reduce the number of fuzzy rules.
Fuzzification: It is used to convert inputs i.e. crisp numbers into fuzzy sets. Crisp inputs are the exact inputs that are measured by sensors and passed into the control system for processing, such as temperature, pressure, rpm’s, etc.
Inference Engine: It determines the matching degree of the current fuzzy input with respect to each rule and then decides which rules are to be fired according to the input field. After that, the fired rules are combined to form the control actions.
De fuzzification: It is used to convert the fuzzy sets obtained by the inference engine into a crisp value. There are several de fuzzification methods available and that are best-suited used with a specific expert system to reduce the error.
Fuzzy Algorithm
A fuzzy logic algorithm helps to solve a problem after considering all available data. Then it takes the best possible decision for the given input. The FL method provides the way of decision making in a human which considers all the possibilities between digital values T and F.
A fuzzy algorithm is applicable to various fields, from control theory to AI. It was designed to allow the computer to determine the dissimilarity among data which is neither true nor false. Something similar to the process of human reasoning. Like a little dark, some brightness, etc.
Fuzzy Control System
A control system is an arrangement of physical components that are designed to alter another physical system so that this system exhibits certain desired characteristics. Given below are some reasons for using Fuzzy Logic in Control Systems −
While applying traditional control, we need to know about the model and the objective function formulated in precise terms. This makes it very difficult to apply in most cases.
By applying fuzzy logic for control, we can utilize human expertise and experience for designing a controller.
The fuzzy control rules basically consist of the IF-THEN rules, and they can be best utilized in designing a controller.
The Architecture of Fuzzy Logic Control
Given diagram below represents the architecture of Fuzzy Logic Control (FLC).
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Major Components of FLC
Followings are the major components of the FLC architecture−
Fuzzifier − The role of the fuzzifier is to convert the crisp input values into fuzzy values.
Fuzzy Knowledge Base − It stores the knowledge about all the input and output fuzzy relationships. It also has the membership function which determines the input variables to the fuzzy rule base and the output variables to the plant that are under control.
Fuzzy Rule Base − It stores the knowledge about the operation of the process of the domain.
Inference Engine − It acts as a kernel of an FLC system. Basically, it simulates human decision making by performing approximate reasoning.
De fuzzifier − The role of fuzzified is to convert the fuzzy values into crisp values getting from the fuzzy inference engine.
Fuzzy Set
Fuzzy logic is based on this theory, which is a generalisation of the classical theory of sets (i.e., crisp set) that was introduced by Zadeh in 1965.
A fuzzy set is a collection of values that exist between 0 and 1. We can denote or represent fuzzy sets by the tilde (~) character. In the fuzzy set, partial membership also exists. This theory was released as an extension of classical set theory.
Mathematically, this theory is denoted as a fuzzy set (Ã) is a pair of U and M, where U is the Universe of discourse and M is the membership function that takes on values in the interval [ 0, 1 ]. The universe of discourse (U) is also denoted by the symbol Ω or X
\[\tilde{A}= \left \{ (x,\mu _{\tilde{A}}(x)) x\in X\right \}\]
Conclusion
As we have discussed, the term fuzzy means things that are not very clear or vague. It is flexible and easy to implement machine learning technique. It should not be used when you can use common sense. We have also discussed architecture that consists of four main parts: Rule Basse, Fuzzification, Inference Engine and De fuzzification. Fuzzy logic takes truth degrees as a mathematical basis, which is based on the model of the vagueness while probability is a mathematical model of ignorance
FAQs on Fuzzy Logic in Mathematics: A Complete Guide
1. What is fuzzy logic in mathematics and how does it differ from traditional Boolean logic?
Fuzzy logic is a mathematical approach that deals with reasoning that is approximate rather than precise. Unlike traditional Boolean logic, which is based on two distinct values (true/false or 1/0), fuzzy logic allows for degrees of truth. This means a statement can be partially true and partially false, represented by a value between 0 and 1. It is designed to model the vagueness and ambiguity inherent in human language and decision-making.
2. How does fuzzy logic differ from probability theory?
While both use values between 0 and 1, they represent different concepts. Fuzzy logic measures the degree of vagueness or ambiguity of an event (e.g., how 'tall' is a person?), where an element can partially belong to a set. In contrast, probability measures the likelihood of a future event occurring (e.g., the chance of it raining tomorrow?). Fuzzy logic is about imprecise definitions, whereas probability is about the chance of a precise outcome.
3. What are the essential components of a fuzzy logic system?
A typical fuzzy logic system consists of four main components:
Fuzzifier: Converts crisp, numerical input data (like temperature in Celsius) into fuzzy sets (like 'cold', 'warm', 'hot').
Rule Base: Contains a set of 'IF-THEN' rules provided by an expert to govern the decision-making process.
Inference Engine: Simulates human reasoning by applying the fuzzy rules to the fuzzy inputs to derive fuzzy outputs.
Defuzzifier: Converts the fuzzy output back into a crisp, numerical value that can be used to control a system (e.g., setting a fan speed).
4. What are some real-world examples of fuzzy logic in action?
Fuzzy logic is widely used in consumer electronics and control systems to handle complex, real-time decisions. Common examples include:
Washing Machines: Adjusting the wash cycle, water level, and time based on the load size and dirtiness of the clothes.
Air Conditioners: Optimising cooling by adjusting the compressor speed based on the current room temperature and the desired temperature.
Anti-lock Braking Systems (ABS): Controlling braking pressure to prevent skidding by interpreting car speed and wheel rotation.
Video Games: Controlling the behaviour of non-player characters (NPCs) to make them act more realistically and less predictably.
5. Why is fuzzy logic often described as being closer to human reasoning?
Fuzzy logic is considered closer to human reasoning because people naturally think in imprecise terms. We make decisions based on concepts like 'a bit too fast', 'slightly warm', or 'very close' rather than exact numerical values. Traditional Boolean logic is too rigid to capture this nuance. Fuzzy logic provides a mathematical framework to work with these vague linguistic variables, mimicking how humans reason and make judgements in complex or uncertain situations.
6. What is a membership function and what is its importance in a fuzzy set?
A membership function is a core concept in fuzzy logic that mathematically defines a fuzzy set. It is a curve or graph that assigns a membership value (a degree of truth between 0 and 1) to each element in the input space. Its importance lies in its ability to quantify a vague concept. For example, for the fuzzy set 'tall', the membership function would assign a value of 0 to someone 5'0" tall, 0.7 to someone 6'0" tall, and 1 to someone 6'8" tall, thus defining 'tallness' across a range of heights.
7. What are the key properties of a fuzzy set?
A fuzzy set is defined by several key properties that distinguish it from a classical (crisp) set:
It is a set whose elements have degrees of membership, a value between 0 and 1.
An element can have partial membership in multiple fuzzy sets within the same universe of discourse.
The degree of membership is not a probability; it represents the degree of truth or belongingness to an imprecisely defined category.
8. What are the main advantages and limitations of fuzzy logic systems?
Advantages:
Fuzzy logic systems are robust and tolerant of imprecise or incomplete input data.
They can model complex, non-linear functions that are difficult to represent with traditional mathematics.
The logic is based on natural language, making the rules and systems easier to understand and modify.
Limitations:
There is no single, systematic method for designing a fuzzy system; it often relies on expert knowledge and trial-and-error.
Verifying and validating a fuzzy system can be more complex than for traditional control systems.
The solutions may not be as precise as those from purely mathematical or data-driven models.
