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### Question 7:
**Given**: \( X = \{3, 4, 8\} \)
**Find**: \( X^2 \) and represent it using both:
1. Arrow diagram
2. Cartesian diagram

- \( X^2 \) refers to the Cartesian product \( X \times X \). This means all ordered pairs where both elements are from set \( X \).

  \[
  X \times X = \{(3, 3), (3, 4), (3, 8), (4, 3), (4, 4), (4, 8), (8, 3), (8, 4), (8, 8)\}
  \]

For representation:
1. **Arrow diagram**: This would involve drawing arrows between elements of \( X \), connecting each element to itself and all others.
2. **Cartesian diagram**: Represent the ordered pairs as points on a Cartesian plane where both axes are labeled with the elements of \( X \).

---

### Question 8:
**Given**:
- \( X = \{3, 4\} \)
- \( Y = \{4, 5\} \)
- \( Z = \{6, 5\} \)

**Find**: 
1. \( X \times (Y \cap Z) \)
   - First, find \( Y \cap Z \) (intersection of \( Y \) and \( Z \)):
     \[
     Y \cap Z = \{5\}
     \]
   - Now, \( X \times (Y \cap Z) \) means the Cartesian product of \( X \) and \( \{5\} \):
     \[
     X \times \{5\} = \{(3, 5), (4, 5)\}
     \]

2. \( (X - Y) \times Z \)
   - First, find \( X - Y \) (difference between \( X \) and \( Y \)):
     \[
     X - Y = \{3\}
     \]
   - Now, \( (X - Y) \times Z \) means the Cartesian product of \( \{3\} \) and \( Z \):
     \[
     \{3\} \times \{6, 5\} = \{(3, 6), (3, 5)\}
     \]

3. \( (X - Y) \times (Y - Z) \)
   - First, find \( Y - Z \) (difference between \( Y \) and \( Z \)):
     \[
     Y - Z = \{4\}
     \]
   - Now, \( (X - Y) \times (Y - Z) \) means the Cartesian product of \( \{3\} \) and \( \{4\} \):
     \[
     \{3\} \times \{4\} = \{(3, 4)\}
     \]

---

### Question 9:
**Given**:
- \( X = \{1, 2\} \)
- \( Y = \{3, 4, 5\} \)

**Find**: \( X \times Y \) and represent it using:
1. Arrow diagram
2. Cartesian diagram

- \( X \times Y \) means the Cartesian product of \( X \) and \( Y \):
  \[
  X \times Y = \{(1, 3), (1, 4), (1, 5), (2, 3), (2, 4), (2, 5)\}
  \]

For representation:
1. **Arrow diagram**: Draw arrows connecting each element of \( X \) to each element of \( Y \).
2. **Cartesian diagram**: Represent these ordered pairs on a Cartesian plane with \( X \) on one axis and \( Y \) on the other.

---

### Question 10:
**Given**: \( X \times Y = \{(2, 3), (2, 2), (2, 4)\} \)
Find:
1. \( X \cap Y \): Find the intersection of \( X \) and \( Y \), given that the elements of the pairs come from the sets \( X \) and \( Y \). Based on the Cartesian pairs, it seems that \( X = \{2\} \) and \( Y = \{2, 3, 4\} \), so:
   \[
   X \cap Y = \{2\}
   \]

2. \( X \times (X \cap Y) \): Now, this is the Cartesian product of \( X \) and \( X \cap Y \). Since \( X = \{2\} \) and \( X \cap Y = \{2\} \), we have:
   \[
   X \times (X \cap Y) = \{(2, 2)\}
   \]

---

Would you like further details on any specific part or help with the diagrams?  
Here are five questions related to this explanation:
1. How do we represent a Cartesian product using an arrow diagram?
2. What are the key differences between a Cartesian diagram and an arrow diagram?
3. How do we find intersections and differences between sets?
4. What is the importance of set operations in Cartesian products?
5. How can the Cartesian product be applied in real-life scenarios like computer science or relations?

**Tip**: In a Cartesian product, the order of elements in the pairs matters. For instance, \( (1, 2) \) is different from \( (2, 1) \).

Solve the problems related to set theory and Cartesian products from the image. Questions include finding X^2 for a set, performing Cartesian products, and representing them with arrow or Cartesian diagrams.

Solve problems related to set theory, including Cartesian products, intersections, and differences. Represent the Cartesian products with arrow and Cartesian diagrams. Suitable for students in Grades 9-11.

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