# Difference between Fuzzy Set and Crisp Set

Fuzzy set and Crisp set are two different set theories. The crisp set utilizes the bi-valued logic. In contrast, the fuzzy set utilizes the infinite-valued logic. Crisp sets have elements with binary membership, whereas fuzzy sets have components with degrees of membership.

In this article, you will learn about the difference between Fuzzy and Crisp sets. But before discussing the differences, you must know about the Fuzzy set and Crisp set with their advantages and disadvantages.

## What is Fuzzy Set?

A fuzzy set is a collection of components with varying degrees of membership in the set. The term "fuzzy" relates to ambiguity; that is, the transition between different degrees of membership means that the boundaries of the fuzzy sets are uncertain. As a result, the membership of the universe's items in the set is compared to a function for detecting uncertainty and ambiguity.

The fuzzy set theory was originally proposed by computer scientist Lotfi A. Zadeh in the year of 1965. After that, various theoretical development in a related topic occurred. In the past, dual logic-based crisp sets were used in formal reasoning and computers to represent solutions in one of two forms, like "true" or "false" and "yes" or "no".

### Fuzzy Logic

Fuzzy logic incorporates approximate human reasoning capabilities to be used in knowledge-based systems. The fuzzy logic theory gives a mathematical way of apprehending ambiguities in the human cognitive process, such as thinking and reasoning, and it may also deal with the issue of lexical imprecision and uncertainty.

1. It is adaptable and can accommodate changes.
2. The logic is robust and straightforward, and it may be adjusted to meet user needs.
3. This logic system has a simple structure and is easy to build.
4. The fuzzy logic systems are simple to build.
5. It can process multiple sorts of inputs at once and make accurate decisions utilizing precise functions.

1. These are not frequently used because of faulty data gathering.
2. Its controller relies entirely on expertise and human knowledge. These controllers are incapable of distinguishing between machine learning and neural networks.
3. The system's efficiency is low since it relies heavily on faulty inputs.
4. The fuzzy logic control system must be upgraded on a regular basis.

## What is Crisp Set?

The crisp set is a group of objects (say U) that have the similar countability and finiteness qualities. A crisp set 'A' is a collection of items over the universal set U, where a random element can or cannot be a part of A. It means that there are only two chances: either the element belongs to set A or does not belong to set A. It is capable of performing operations such as union, intersection, complement, and difference. The crisp set displays various features, including idempotency, commutativity, transitivity, distributivity, associativity, identity, and involution. However, fuzzy sets also share the same functionalities.

### Crisp Logic

The standard method of knowledge representation doesn't offer a suitable means to interpret ambiguous and non-categorical data. Since its operations are based on classical probability theory and first-order logic, in another sense, it is unable to portray human intelligence.

### Features of Crisp Set

There are various features of the Crisp Set. Some main features of Crisp Set are as follows:

1. An element's boolean condition of obedience in a crisp set denotes either membership or not.
2. A set in which each element has degree one of membership and each element in the complement of this set has degree zero of membership.
3. A typical set is where an element is either a member or not of the set.
4. A set that is specified using a characteristic function that gives each element of the universe a value of either 0 or 1, distinguishing between elements that are members of the crisp set under consideration and those that are not.

## Key Differences between Fuzzy Set and Crisp Set

There are various key differences between Fuzzy Set and Crisp Set. Some main differences between Fuzzy Set and Crisp Set are as follows:

1. The indeterminate limits of a fuzzy set define it, and there is doubt about the set's boundaries. In contrast, a crisp set is characterized by crisp boundaries and has the specific location of the set boundaries.
2. The fuzzy set adheres to the logic of infinite values. In contrast, a crisp set is built on bi-valued logic.
3. A fuzzy set has partial membership, which implies it ranges from true to false, yes to no, and 0 to 1. On the other hand, the crisp set is a full membership which means completely true/false, 0/1.
4. A fuzzy set defines the degree to which anything is true. On the other hand, the Crisp set is also known as a classical set.
5. The fuzzy set specifies a number between 0 and 1, which includes both 0 and 1. On the other hand, the crisp set specifies the value as either 0 or 1.
6. The fuzzy set is utilized in the fuzzy controller. On the other hand, the Crisp set is employed for digital design.
7. Crisp and fuzzy set theory have different apps, but both are geared toward the construction of efficient expert systems.
8. A fuzzy set indicates incomplete membership. On the other hand, the crisp set displays complete membership.

Here, you will learn the head-to-head comparisons between Fuzzy Set and Crisp Set. The main differences between Fuzzy Set and Crisp Set are as follows:

Features Fuzzy Set Crisp Set
Basic It is prescribed by vague or ambiguous properties. It is defined by precise and specific characteristics.
Definition It is a set of components with different membership degrees in the set. It is a set of objects that have the same countability and finiteness qualities.
Applications It is commonly utilized in fuzzy controllers. It is commonly utilized in digital design.
Membership It shows incomplete membership. It shows the complete membership.
Logic It follows the infinite-valued logic. It follows the bi-valued logic.
Value It specifies a number between 0 and 1, which includes both 0 and 1. It specifies the value as either 0 or 1.
Degree It defines the degree to which anything is true. It is also referred to as a classical set.

## Conclusion

The fuzzy set theory is designed to bring vagueness and imprecision to attempt to replicate the human brain in AI. Its importance in the field of expert systems is growing by the day. On the other hand, the Crisp set theory was particularly useful as the starting notion for modelling digital and expert systems based on binary logic.

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