Fuzzy Set Theory is a mathematical concept that deals with the representation of uncertainty or vagueness in data. Unlike the conventional set theory, where an element either belongs or doesn't belong to a set, Fuzzy Set Theory allows partial membership, meaning an element can belong to a set in a partial or fuzzy manner.
| # | Fuzzy Sets | Crisp Sets |
|---|---|---|
| 1 | Elements have degrees of membership, ranging from 0 to 1 | Elements either belong to the set or do not belong |
| Example | The temperature of a room can be a fuzzy concept. We can define a set "comfortable temperature" with elements having different degrees of membership. | The set of even numbers, where the elements are well-defined and either belong to the set or do not. |
| 2 | Membership functions represent the degree of membership | No membership function |
| Example | The membership function for the set "old age" can have elements with different degrees of membership based on a person's age. | The set of prime numbers, where there is no need for a membership function since the set is well-defined. |
| 3 | Fuzzy sets enable a range of possibilities between being a member or not being a member. | No clear distinction between being a member or not being a member of a set. |
| Example | A person can partially belong to the set "tall people" based on their height. | A person either belongs to the set "voters" or does not, with no gradation in between. |
| 4 | Fuzzy sets are used to represent uncertainty and vagueness | Crisp sets are used to represent exact, well-defined concepts |
| Example | The set "good music" is a fuzzy concept since what is considered good varies based on individual preferences. | The set of integers, where there is a well-defined rule for what constitutes an integer. |
| 5 | Fuzzy sets have infinite boundaries | Crisp sets have finite boundaries |
| Example | The set "nearby cities" has infinite boundaries since what is considered nearby can vary based on distance or time. | The set of animals, where the boundaries are well-defined and limited to specific species. |
| 6 | Fuzzy sets can overlap with each other | Crisp sets cannot overlap |
| Example | The sets "tall people" and "thin people" can overlap since a person can be both tall and thin. | The set of even numbers and the set of odd numbers cannot overlap since they are mutually exclusive. |
| 7 | Fuzzy sets can be combined using operations such as fuzzy intersection and fuzzy union | Crisp sets can be combined using operations such as intersection and union |
| Example | The sets "young people" and "students" can be combined using fuzzy intersection to create the set of "young students". | The sets of even numbers and multiples of 3 can be combined using intersection to create the set of even multiples of 3. |
| 8 | Fuzzy sets can be used to model imprecise phenomena such as human emotions and linguistic concepts | Crisp sets are better suited for modeling precise phenomena such as physical objects and mathematical concepts |
| Example | The set "happiness" is a fuzzy concept since it can vary based on individual interpretations. | The set of integers, which is well-defined and precise. |
| 9 | Fuzzy sets require more complex mathematical operations than crisp sets | Crisp sets require simpler mathematical operations than fuzzy sets |
| Example | Fuzzy sets require operations such as fuzzy complement and fuzzy composition to fully represent the complexities of uncertainty and vagueness. These operations involve complex mathematical equations that take into account the degree of membership and the shape of the membership function. | Crisp sets require simple operations such as addition and subtraction to manipulate the elements of the set. These operations are straightforward and do not require complex mathematical equations. |
| 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. |
In mathematics, a relation is a set of ordered pairs that relate elements of one set to elements of another set. Crisp relations are relations where each element in the set is either fully in the relation or fully out of the relation. For example, the relation "less than" between the numbers 1 and 5(1<x<5) is a crisp relation since any number that is less than 1 or greater than 5 is not in the relation, and any number between 1 and 5 is in the relation.
On the other hand, fuzzy relations are relations where the degree of membership of each element in the relation can vary between 0 and 1, indicating the degree of similarity or relevance between the elements. For example, the relation "is tall" between humans is a fuzzy relation, since the degree of membership of a person being tall can vary from 0 to 1, depending on their height.
One of the applications of fuzzy relations is in the field of expert systems, where they can be used to model the knowledge and expertise of human experts in a fuzzy rule-based system. For example, in a medical diagnosis system, fuzzy relations can be used to model the degree of correlation between symptoms and diseases, based on the expertise of doctors.
In summary, while crisp relations have binary membership, fuzzy relations allow for degrees of membership, which can help in modeling the complexities of real-world situations where elements may have varying degrees of relevance or similarity.
Fuzzy systems are a type of artificial intelligence system that use fuzzy logic to represent and manipulate uncertain or vague information. In contrast, crisp logic deals with information that is either true or false, with no in-between. Let's take a look at the differences between crisp and fuzzy logic using an example.
Suppose we want to classify people into two groups: tall and short. Using crisp logic, we would define a threshold height and consider anyone taller than the threshold as tall, and anyone shorter as short. For example, we might set the threshold height to 6 feet, so anyone taller than 6 feet would be classified as tall, and anyone 6 feet or shorter would be classified as short. This approach works well for well-defined categories, but it doesn't account for the fact that height can be a matter of degree, and different people might have different opinions on what constitutes "tall" or "short".
Fuzzy logic, on the other hand, allows for degrees of membership, which can better represent the vagueness or uncertainty of real-world situations. In fuzzy logic, we would use membership functions to represent the degree of membership of an element in a set. For example, we could define a membership function for the set "tall" that varies from 0 to 1, where 0 represents not tall at all, 1 represents completely tall, and values in between represent degrees of tallness. The membership function might look something like this:
$$
\begin{equation} \mu_{\text{Tall}}(x) = \begin{cases} 0.2 & \text{if } x \leq 5 \text{ feet} \\ \frac{x-5}{6-5} & \text{if } 5 \leq x \leq 6 \text{ feet} \\ 1 & \text{if } x \geq 6 \text{ feet} \end{cases} \end{equation}
$$