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Let's go through each problem in detail.

### Problem 1
You are asked to write the membership function of a fuzzy set \( A = \{0, 4, 7, 10\} \) with a trapezoidal shape. A trapezoidal membership function \( \mu_A(x) \) is often defined as:
\[
\mu_A(x) = \begin{cases} 
0, & x \leq a \\
\frac{x - a}{b - a}, & a < x \leq b \\
1, & b < x \leq c \\
\frac{d - x}{d - c}, & c < x \leq d \\
0, & x \geq d 
\end{cases}
\]
For the given points, assume the trapezoidal function parameters are \( a = 0 \), \( b = 4 \), \( c = 7 \), and \( d = 10 \).

### Problem 2
Here, you need to find the intersection of two fuzzy sets \( A \) and \( B \), defined by triangular membership functions:
- \( A = \{1, 2, 4\} \)
- \( B = \{2, 4, 6\} \)

The triangular membership function is generally defined as:
\[
\mu(x) = \begin{cases} 
0, & x \leq a \\
\frac{x - a}{b - a}, & a < x \leq b \\
\frac{c - x}{c - b}, & b < x < c \\
0, & x \geq c 
\end{cases}
\]
For the intersection \( C \), you would take the minimum of the membership values of \( A \) and \( B \) at each point.

### Problem 3
This task asks for the logical product (intersection) of the fuzzy sets \( HM \) \( A \) and \( B \). The logical product in fuzzy logic is typically the minimum of the membership values for corresponding elements in \( A \) and \( B \).

### Problem 4
You are asked to create two finite fuzzy sets on the real line and find the Chebyshev distance between them. The Chebyshev distance between two sets \( X \) and \( Y \) (with points \( x_i \) in \( X \) and \( y_i \) in \( Y \)) is:
\[
D_\infty(X, Y) = \max |x_i - y_i|
\]

### Problem 5
Here, you are asked to provide an example of a tabular representation of a fuzzy relation on \( X \times X \), where \( X \) is the finite set of natural numbers in the interval \([3, 10]\). The table should show values that meet antisymmetry and antireflexivity conditions.

### Problem 6
The task involves defining a normalized fuzzy set on 11 elements with an alpha-cut at \( \alpha = 0.1, 0.3, 0.7 \). This requires creating a fuzzy set where the membership values are scaled such that the maximum value is 1, and identifying the elements that satisfy each alpha-cut condition.

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Do you want further details on any specific step? Let me know if you have questions on fuzzy set membership functions or fuzzy set operations.

### Further Questions
1. What is the general form of a trapezoidal membership function and how is it used?
2. How do you calculate the intersection of two triangular fuzzy sets?
3. What is the logical product in fuzzy set theory, and how does it relate to membership functions?
4. How is Chebyshev distance used in the context of fuzzy sets on a real line?
5. How can you construct a tabular representation for antisymmetric and antireflexive fuzzy relations?

#### Tip
When dealing with fuzzy sets, remember that operations such as union, intersection, and complement are often defined differently from classical set theory, typically using functions like max, min, and 1-complement.

Write the membership function of a fuzzy set A with a trapezoidal shape (A = {0, 4, 7, 10}). Calculate the intersection of fuzzy sets A and B with triangular functions (A = {1, 2, 4}, B = {2, 4, 6}). Find the logical product of sets A and B. Define two finite fuzzy sets on the real line and find their Chebyshev distance. Provide an example of a tabular representation of a fuzzy relation with specific conditions. Create a normalized fuzzy set of 11 elements and specify its alpha-cuts.

Explore fuzzy set theory with problems on trapezoidal and triangular membership functions, intersection and logical product of fuzzy sets, Chebyshev distance, and tabular representations of fuzzy relations. Includes solutions for normalized fuzzy sets and alpha-cuts.

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