https://file.aiquickdraw.com/mathbot/file/1727043736600mosib94m.jpg

The image presents two fuzzy sets, \( A \) and \( B \), based on a universal set \( U = \{u_1, u_2, \dots, u_8\} \), shown as a diagram. The plot represents the membership functions of these fuzzy sets, where the vertical axis \( \mu(u) \) represents the degree of membership, and the horizontal axis represents the elements \( u_1 \) to \( u_8 \).

The red solid line corresponds to fuzzy set \( A \), and the blue dashed line corresponds to fuzzy set \( B \). The task at the bottom is to compute the distance between \( A \) and \( B \) using the linear metric \( d^L(A, B) \).

### Steps to compute the linear metric distance:
In fuzzy set theory, the linear metric (or Manhattan distance) between two fuzzy sets \( A \) and \( B \) is given by:

\[
d^L(A, B) = \sum_{i=1}^{n} |\mu_A(u_i) - \mu_B(u_i)|
\]

Where:
- \( \mu_A(u_i) \) and \( \mu_B(u_i) \) are the membership values of elements \( u_i \) in fuzzy sets \( A \) and \( B \), respectively.

From the graph, you need to extract the membership values of both sets \( A \) and \( B \) for each element \( u_1 \) through \( u_8 \), and then calculate the sum of the absolute differences.

Would you like me to extract the specific membership values from the graph and calculate the distance for you?

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Here are 5 follow-up questions related to the task:
1. What are the typical applications of fuzzy sets in real-world problems?
2. How is the linear metric different from other metrics, such as the Euclidean metric, in fuzzy set theory?
3. Can you explain the concept of a membership function in more detail?
4. How does the choice of distance metric affect the results when comparing fuzzy sets?
5. What other types of fuzzy set operations can be performed besides distance calculation?

**Tip:** The linear metric, also known as the Manhattan distance, emphasizes the total difference in membership values across all elements, making it useful for detecting overall discrepancies between two fuzzy sets.

На рисунке ниже приведены диаграммы 2-х нечётких множеств, построенных на основе универсального множества U = {u1,...,u8}. Рассчитать расстояние в линейной метрике между нечёткими множествами A и B.

Learn how to calculate the linear metric (Manhattan distance) between two fuzzy sets, A and B, using their membership functions. This example provides a detailed explanation and steps to compute the distance between two fuzzy sets based on universal set U = {u1,...,u8}.

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