https://file.aiquickdraw.com/mathbot/file/17288488561733u3y6ya9.jpg

Let's begin by analyzing the questions in the image you provided. I'll go step by step:

### 1. Write the set-builder form of \( A \):
Given:
\[ A = \left\{1, \frac{1}{4}, \frac{1}{9}, \frac{1}{16}, \frac{1}{25}\right\} \]

The set-builder form will describe the rule that generates these elements:
\[ A = \left\{ x : x = \frac{1}{n^2}, \text{ where } n \in \mathbb{N} \text{ and } 1 \leq n \leq 5 \right\} \]

### 2. Set Operations: Factors of 12 and 18
Given sets:
- \( A = \left\{ x : x \text{ is a factor of 12} \right\} \)
- \( B = \left\{ x : x \text{ is a factor of 18} \right\} \)

#### Finding the elements:
- Factors of 12: \( A = \{1, 2, 3, 4, 6, 12\} \)
- Factors of 18: \( B = \{1, 2, 3, 6, 9, 18\} \)

Now solve:
- \( A \cup B = \{1, 2, 3, 4, 6, 9, 12, 18\} \)
- \( A \cap B = \{1, 2, 3, 6\} \)
- \( A - B = \{4, 12\} \)
- \( B - A = \{9, 18\} \)

### 3. Set Operations: Factors of 24 and 36
Given sets:
- \( A = \left\{ x : x \text{ is a factor of 24} \right\} \)
- \( B = \left\{ x : x \text{ is a factor of 36} \right\} \)

#### Finding the elements:
- Factors of 24: \( A = \{1, 2, 3, 4, 6, 8, 12, 24\} \)
- Factors of 36: \( B = \{1, 2, 3, 4, 6, 9, 12, 18, 36\} \)

Now solve:
- \( A \cup B = \{1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36\} \)
- \( A \cap B = \{1, 2, 3, 4, 6, 12\} \)
- \( A - B = \{8, 24\} \)
- \( B - A = \{9, 18, 36\} \)

### 4. Set Operations: Prime Numbers and Natural Numbers Less Than 15
Given sets:
- \( A = \left\{ x : x \text{ is a prime number and } x \leq 10 \right\} = \{2, 3, 5, 7\} \)
- \( B = \left\{ x : x \text{ is an odd natural number and } x < 15 \right\} = \{1, 3, 5, 7, 9, 11, 13\} \)

Now solve:
- \( A \cup B = \{1, 2, 3, 5, 7, 9, 11, 13\} \)
- \( A \cap B = \{3, 5, 7\} \)
- \( A - B = \{2\} \)
- \( B - A = \{1, 9, 11, 13\} \)

### 5. Set Operations: Perfect Squares and Perfect Cubes
Given sets:
- \( A = \left\{ x : x \text{ is a perfect square and } x \leq 100 \right\} = \{1, 4, 9, 16, 25, 36, 49, 64, 81, 100\} \)
- \( B = \left\{ x : x \text{ is a perfect cube and } x \leq 150 \right\} = \{1, 8, 27, 64, 125\} \)

Now solve:
- \( A \cup B = \{1, 4, 8, 9, 16, 25, 27, 36, 49, 64, 81, 100, 125\} \)
- \( A \cap B = \{1, 64\} \)
- \( A - B = \{4, 9, 16, 25, 36, 49, 81, 100\} \)
- \( B - A = \{8, 27, 125\} \)

### 6. Set Operations: Arithmetic Sequences
Given sets:
- \( A = \left\{ x : x = 2n + 1, n \in \mathbb{N}, n \leq 5 \right\} = \{3, 5, 7, 9, 11\} \)
- \( B = \left\{ x : x = 3n - 1, n \in \mathbb{N}, n \leq 5 \right\} = \{2, 5, 8, 11, 14\} \)

Now solve:
- \( A \cup B = \{2, 3, 5, 7, 8, 9, 11, 14\} \)
- \( A \cap B = \{5, 11\} \)
- \( A - B = \{3, 7, 9\} \)
- \( B - A = \{2, 8, 14\} \)

These are the steps for the first six problems. Would you like details on the remaining ones or specific questions? Here's a list of five related questions to expand on:

1. What is the significance of the intersection of two sets in real-world problems?
2. Can the union of two sets ever be smaller than either set? Why or why not?
3. What are some examples of real-world applications for set differences?
4. How can Venn diagrams help in solving problems involving more than two sets?
5. What are the differences between perfect squares and perfect cubes in terms of their occurrence in sets?

**Tip:** Always ensure to double-check set operations when dealing with intersections and differences, as missing elements can lead to incorrect results.

Solve set theory problems involving union, intersection, and difference for factors of numbers, prime numbers, perfect squares, and cubes, and arithmetic sequences.

This problem involves set operations like union, intersection, and difference. Sets include factors of numbers (12, 18, 24, 36), prime numbers, perfect squares, cubes, and arithmetic sequences. Suitable for high school students.

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