https://file.aiquickdraw.com/mathbot/file/1729193884231np2w57gm.jpg

The question in the image involves set theory and algebraic notation. Let's break it down step by step:

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

### The tasks are:
1. \( X \times (Y \cap Z) \)
2. \( Y^2 \)
3. \( n(X^2) \)

---

### Solution:

#### 1. \( X \times (Y \cap Z) \)

- **Intersection of Y and Z:**
  \[
  Y \cap Z = \{4, 5\} \cap \{5, 6\} = \{5\}
  \]
- **Cartesian product:**
  \[
  X \times \{5\} = \{(3, 5), (4, 5)\}
  \]

#### 2. \( Y^2 \)

- This notation \( Y^2 \) seems unclear, but in set theory, it could mean the Cartesian product \( Y \times Y \). If this is the case:
  \[
  Y \times Y = \{(4, 4), (4, 5), (5, 4), (5, 5)\}
  \]
  So, \( Y^2 = Y \times Y \).

#### 3. \( n(X^2) \)

- \( X^2 \) probably refers to \( X \times X \), the Cartesian product of \( X \) with itself:
  \[
  X \times X = \{(3, 3), (3, 4), (4, 3), (4, 4)\}
  \]
- **Number of elements in \( X^2 \):**
  \[
  n(X^2) = 4
  \]

---

Would you like further details on any part of this solution?

Here are 5 related questions to expand your understanding:
1. What is the union of sets \( X \), \( Y \), and \( Z \)?
2. How does the Cartesian product differ from the union of sets?
3. Can you compute \( (Y \cup Z) \times X \)?
4. What is the power set of \( X \)?
5. How would the intersection of \( X \) and \( Z \) be computed?

**Tip:** Cartesian products produce pairs of elements from two sets, making the result much larger than simple set operations like union or intersection.

If X = {3, 4}, Y = {4, 5}, and Z = {5, 6}, find:
1. X × (Y ∩ Z)
2. Y^2
3. n(X^2)

This problem explores set theory by calculating Cartesian products and intersections of sets. You'll work with the sets X = {3, 4}, Y = {4, 5}, and Z = {5, 6} to solve tasks like finding X × (Y ∩ Z), Y^2, and n(X^2).

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