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

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

Would you like me to solve more problems or provide further clarifications?

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.