https://file.aiquickdraw.com/mathbot/file/1729437581035r65ebrkx.jpg

I have displayed the image for analysis. Let's go through and solve the problems presented in the image:

### Problem 1:
\[
\text{If: } (x^3, y + 3) = \left( 1, \sqrt{4} \right), \text{ then } X - Y = \ldots
\]
\[
\text{Solution: } x^3 = 1 \Rightarrow x = 1 \quad \text{and} \quad y + 3 = 2 \Rightarrow y = -1.
\]
Thus, \( X - Y = 1 - (-1) = 2 \).
**Answer: (b) 2**

---

### Problem 2:
\[
\text{If: } (a + 1, 5) = (2, b - 1), \text{ then: } 2a + b = \ldots
\]
\[
\text{Solution: } a + 1 = 2 \Rightarrow a = 1 \quad \text{and} \quad 5 = b - 1 \Rightarrow b = 6.
\]
Thus, \( 2a + b = 2(1) + 6 = 8 \).
**Answer: (d) 8**

---

### Problem 3:
\[
\text{If: } X = \{3\}, \text{ then: } X^2 = \ldots
\]
\[
\text{Solution: } X^2 = \{x^2 \mid x \in X\} = \{3^2\} = \{9\}.
\]
**Answer: (c) \{9\}**

---

### Problem 4:
\[
\text{If: } X = \{5\}, Y = \varnothing, \text{ then: } X \setminus Y = \ldots
\]
\[
\text{Solution: } X \setminus Y = X = \{5\}, \quad \text{since subtracting an empty set leaves the original set}.
\]
**Answer: (a) 1**

---

### Problem 5:
\[
\text{If: } X = \{3\}, Y = \{4, 5\}, \text{ then } X \cap Y = \ldots
\]
\[
\text{Solution: } X \cap Y = \varnothing \quad \text{(no common elements)}.
\]
**Answer: (d) \varnothing**

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Let me know if you'd like detailed explanations for any of these, and I can continue solving the remaining problems from the image.

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### Follow-up Questions:
1. What is the importance of set operations in algebra?
2. How can you generalize the idea of finding \(X - Y\) for different sets?
3. In problem 3, how does squaring a set operate?
4. Can you provide a real-world example of a set difference?
5. How do ordered pairs (as in problem 1) behave in different fields like physics?

#### Tip: Always check if the sets involved are empty, as this affects set operations like union and intersection.

Solve the given mathematical problems related to algebra, set theory, and set operations.

Detailed solutions to algebraic and set theory problems, including set difference, intersection, and squaring operations. Suitable for students in Grades 6-9, these problems enhance understanding of set operations and algebraic manipulation.

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