13. How to perform Max-Min and Max Product Composition between two fuzzy sets by Mahesh Huddar

Summary

This video provides a clear explanation of how to perform Max-Min and Max-Product composition on fuzzy relations. It walks through an example of two given fuzzy relations, R and S, and demonstrates how to compute the new fuzzy relation T using Max-Min composition, followed by Max-Product composition. The process includes calculating membership values for each element in the relations, with detailed steps for each computation. The video aims to clarify these concepts with a practical example, making it easier to understand fuzzy relation composition methods.

Takeaways

• 📊 The video explains how to perform Max-Min and Max-Product compositions on fuzzy relations with examples.
• 📝 Given two fuzzy relations R and S, the goal is to find a new fuzzy relation T after performing these compositions.
• 🔢 For the Max-Min composition, the number of columns in the first relation (R) must match the number of rows in the second relation (S).
• 📈 The result of the Max-Min composition, relation T, will have as many rows as R and as many columns as S.
• 📉 Membership values in the new relation T are calculated by taking the minimum of pairs of values from R and S, followed by taking the maximum of these minima.
• ⚙️ An example is provided where membership values for pairs like X1Z1, X1Z2, and so on are computed using the Max-Min rule.
• 🧮 For Max-Product composition, instead of taking the minimum of values, the product of values is taken, followed by the maximum.
• 🔄 The process for calculating Max-Product is similar to Max-Min, but with multiplication instead of minimum values.
• 💡 The video highlights how to compute membership values for different combinations like X1Z1, X1Z2, etc., for both Max-Min and Max-Product compositions.
• 📚 The tutorial concludes with a final summary of the membership values for relation T after applying both composition methods.

Q & A

What are the two main operations discussed in the video for fuzzy relations?

The two main operations discussed are Max-Min composition and Max-Product composition on fuzzy relations.

What is the basic condition to perform Max-Min or Max-Product composition on two fuzzy relations?

The number of columns in the first fuzzy relation must match the number of rows in the second fuzzy relation.

How is the resulting fuzzy relation 'T' structured after performing Max-Min composition?

The resulting fuzzy relation 'T' will have rows equal to the number of rows in the first fuzzy relation and columns equal to the number of columns in the second fuzzy relation.

What is the process for calculating the membership value for X1 and Z1 in the Max-Min composition?

First, the minimum values between X1 and Y1, and Y1 and Z1 are calculated. Then, the minimum between X1 and Y2, and Y2 and Z1 is calculated. The maximum of these two minimum values becomes the membership value for X1 and Z1.

What is the difference between Max-Min and Max-Product composition?

In Max-Min composition, the minimum of two values is used, whereas in Max-Product composition, the product of two values is used, and then the maximum of those products is selected.

How do you calculate the membership value for X1 and Z2 in the Max-Product composition?

The process involves multiplying X1 and Y1, and Y1 and Z2 to get the first product, and then multiplying X1 and Y2, and Y2 and Z2 to get the second product. The maximum of these two products is the membership value for X1 and Z2.

Why is it necessary for the number of columns in the first fuzzy relation to match the number of rows in the second?

This condition ensures that each element in the first relation has a corresponding element in the second relation, which allows the composition operations to be carried out correctly.

How many membership values need to be calculated for the resulting fuzzy relation 'T' when performing Max-Min composition?

For the resulting fuzzy relation 'T', six membership values need to be calculated, corresponding to all combinations of the rows from the first relation and the columns from the second relation.

What is the final membership value for X1 and Z3 after performing Max-Product composition?

The membership value for X1 and Z3 is calculated by taking the maximum of the products of X1 to Y1 and Y1 to Z3, and X1 to Y2 and Y2 to Z3, which results in a value of 21.

How do Max-Min and Max-Product compositions differ in their application to fuzzy relations?

Max-Min composition selects the maximum of the minimum values between elements, while Max-Product composition selects the maximum of the product values between elements, making Max-Product more sensitive to the degree of membership strength between relations.