What and Why Fuzzy Set ?

Fuzzy set is a natural way to deal with the imprecision. Many real world representation relies on significance rather than precision. Fuzzy logic is the best way to deal with them. Fuzzy set is an extension of crisp set.

Fuzzy Set – What wise men say?

Precision is not truth.

— Henri Matisse

So far as the laws of mathematics refer to reality, they are not certain. And so far as they are certain, they do not refer to reality.

— Albert Einstein

As complexity rises, precise statements lose meaning and meaningful statements lose precision.

— Lotfi Zadeh
Henri Matisse , Albert Einstein and Lotfi Zadeh
Henri Matisse (Left), Albert Einstein (Center) and Lotfi Zadeh (Right)

According to all the above greats, precision is not the everything. Most of the times, we are looking for the significance rather than precision.

For example if some heavy object is approaching to your friend from the top and you say that “hey friend, 1500 kg mass is approaching to you at the speed of 45.3 m/s”. This statement is quite precise but before your friend Interpret it, the object might have fallen on him. 

But if you simply shout “hello friend, lookout” then probably statement is not so precise but it convey quick information to your friend and he may move away from his place. In real world, we may be looking for significance rather than precision. Fuzzy logic helps us to model imprecision into data for many real world problems 

Precision vs. Significance
Precision vs. Significance

History of fuzzy logic

Though fuzzy logic gain popularity after the seminal work of Lofti Zadeh in 1965, the roots of fuzzy logic are very old. The concept of fuzzy logic dates back to the era of Aristotle.

  • Aristotle supported the law of the Excluded Middle, i.e. there is no concept of partial truth. According to him, the proposition can be either true or false, it can not take middle value. This fact can be modelled by crisp set membership, where each element take membership value 0 or 1. The claim of Aristotle was strong base for binary valued logic.
  • Later Heraclitus challenged the claim of Aristotle and claimed that things could be simultaneously True and not True. If we ask someone about Mr. A – if he is good or not? The answer is subjective. He might be good for some people, and he may not be good for others. So the predicate takes both the values simultaneously.
  • Plato – the pupil of Aristotle – laid the foundation for what would become fuzzy logic. He said that the proposition could take values other then the extrema’s, i.e. true (1) and false(0). According to him, predicate could take true and not true value simultaneously as well as the truthiness of statement not necessarily be 0 or 1. This logic suggest that the goodness of Mr. A can very between 0 and 1, rather then 0 or 1. This could be considered as pioneer thought in the direction of multivalued logic.
  • Later, Łukasiewicz described a three-valued logic (True, False, Possible), along with the mathematics to accompany it. The possibility in fact can be correlated with probability of statement being true, which can take any real number between 0 and 1. Hence, it supports the claim of Plato of multi valued logic
  • Lofti Zadeh introduced the notion of an infinite-valued logic in his seminal work ”Fuzzy Sets” where he described the mathematics of fuzzy set theory, and by extension fuzzy logic. Rightly he is being referred as the father of fuzzy logic. He has proposed systematic way of modeling imprecision of the data using fuzzy sets. He also proposed strong mathematical background to support his claim.

What is fuzzy logic

Fuzzy logic is an approach to computing based on “degree of truth” rather than the usual “true or false value” (0 or 1) Boolean logic on which the modern computer is based.

Crisp representation vs. Fuzzy representation
Crisp representation (left) vs. Fuzzy representation (right)

Crisp logic is binary valued logic.  Any element in set has membership value either 0 or 1. Whereas, Fuzzy Logic which is also known as multi-valued has multiple membership values possible. For multi-valued logic, element can take any real value between 0 and 1.

For instance, if you ask Alex and John are good friends? Using crisp representation, the possible outcomes for this question is either 0 or 1. But with fuzzy logic, the possible outcomes could be such as they are extremely good friend, they are good friends, they are not so good friend, or they are not friends at all. This is subjective approach and the membership value of the element is assigned based on the perception or the confidence of the user. For particular case, we have assigned membership value 1 to extremely good, 0.6 to very good, 0.2 to not so good and 0.0 to not at all

Suggested reading: Introduction to crisp set


Universal set (top), crisp set (middle), fuzzy set (bottom)

As discussed in the article on introduction to crisp set, we can consider the class of students as a universal set.  If we ask question “who does have a driving licence?”, – student may or may not have driving licence. Based on that, hte membership value assigned to student will be either 0 or 1

But if we ask the question “who can drive well?” – answer to this question is quite subjective. Based on the skill of student, the membership value of student in particular set will vary from 0 to 1, where 0 indicates no driving skill and 1 indicate the highest level of driving skill. This is how fuzzy representation helps us to capture the uncertainty in the data.

From above examples, we can say that the crisp sets have crisp boundary (element is either inside set or outside set), and fuzzy set is having fuzzy boundary (element can be partially member of set)

Motivation for fuzzy set

There are many motivational factors for fuzzy set. Few of them are listed here.

  • Knowledge in real world can be: inaccurate, unclear, imprecise, indecisive, probabilistic, approximate. Crisp set can not tolerate the imprecision in the data.
  • Human thinking and reasoning include fuzzy nuances. Human mind compares/measure the things relatively, for example, while we touch something, instead of measuring exact value of temperature, we think whether the object is cold, warm, hot or very hot etc. Fuzzy set inherently represents the real data into such linguistic terms.
  • Real world systems should function with vague information. Many times, data might be missing or may not be recorded. Fuzzy logic deals with the range, rather then individual elements, so its easy to handle such vague information too.
  • Fuzzy systems are suitable for uncertain or approximate reasoning, especially for the system with a mathematical model that is difficult to derive.
  • Fuzzy logic allows decision making with estimated values under incomplete or uncertain information.

Watch on YouTube


Advantages of fuzzy set

Fuzzy sets are quite useful in many industrial and hand hold device design due to its simplicity and interpretability. Some of the most prevailing advantages are listed here:

  • Conceptually easy to understand as it is built upon “natural” mathematics
  • Tolerant to imprecise data
  • Universal approximation: It can model arbitrary nonlinear functions
  • Intuitive
  • Based on linguistic terms
  • Convenient way to express expert and common sense knowledge

Limitations of fuzzy set

  • How do we define the membership functions? For the same range of inputs, different person can use different membership function, which results into different result.
  • There is no proper way of learning in fuzzy logic. Membership function is to be chosen from the experience of the domain expert.
  • What if we have membership functions provided from two different people E.g. What a 6’11” Basketball player defines as tall will differ from a 4’10” Gymnast. A guy having height 6.11 may not be considered tall in case of Basketball, but a player with height 4.10 is considered as tall in Gymnast. Thus, there might be ambiguity in interpretation of membership function, that is, domain changes, interpretation also changes.
  • Defuzzification can produce undesired results. Different defuzzification techniques can produced different crisp output for same fuzzy value. The produced crisp value might have large variations in them, this creates lots of ambiguity in selection of defuzzification method.
  • Crisp/precise models can be more efficient and even convenient
  • Membership values begin to move away from expectations when chains of logic are lengthy so this approach is not suitable for many KBS problems (e.g., medical diagnosis)

Probability vs. Fuzziness

People often miss-interpret probability with fuzziness.

Probability describes the uncertainty of an event occurrence.

  • Probability defines frequency of likelihood that an element is in a class
  • There is a 50% chance of an apple being in the refrigerator: If you open the refrigerator 100 times, 50 times you will find the apple in refrigerator

Fuzziness describes event ambiguity.

  • Fuzziness defines similarity of an element to a class
  • There is a half an apple in the refrigerator: If you open the refrigerator 100 times, all 100 times you will find half apple (50%) in refrigerator.

Applications of fuzzy logic

Fuzzy logic covers wide spectrum of applications. Few of them are listed here.

  • Reasoning tool like Fuzzy Logic Controller
  • Automation (Flight Control, Washing Machine, …)
  • Environment Control (Air Conditioner)
  • Clustering using fuzzy logic
  • Fuzzy Mathematical Programming
  • Fuzzy Graph Theory
  • Hybrid systems (ANFIS)

Test your knowledge

  1. What is the fundamental difference between fuzzy set and crisp set?
  2. State the scenario where crisp set is preferred over fuzzy set?
  3. State the scenario where fuzzy set is preferred over crisp set?
  4. Who is known as the father of fuzzy logic?

Please post your answers / queries /feedback in comment box below !

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8 Responses

  1. mohammadc says:

    interesting details and explaination

  2. Parekh Payal says:

    Excellent Sir, Detail topic wrote here that is very informative and furthermore, It had more clear by watching your youtube video.

  3. Himani says:

    This is the best place to learn about fuzzy! I request you to come up with the blog on the topics like Machine Learning.

  4. Smita Mahajani says:

    1 )Crisp set defines the value either 0 or 1 , fuzzy set is between 0 and 1
    2 ) crisp set shoes size
    3 ) fuzzy set fuzzy controller (washing machine)
    4) Zadeh

    • codecrucks says:

      Thanks Smita for posting the answers.
      Crisp sets are preferable when there is only full membership is possible, such as “student is pass or fail” predicate is better represented with crisp set. Where as the fuzzy sets are preferable when variable can take any real value. For example, to measure the hotness of water, pressure of air we can use fuzzy set.

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