The image you uploaded contains mathematical problems involving sets and number lines. Let me walk through each question step-by-step:

2. Represent each of the following sets on a number line:

• a) $\mathbb{R} \setminus \{1\}$

  • This represents all real numbers except $1$. On a number line, this would be a continuous line with an open circle at $1$ (indicating that $1$ is not included).

• b) $\mathbb{R} \setminus \{1, 2\}$

  • This represents all real numbers except $1$ and $2$. Similar to the previous one, it would be a continuous line with open circles at both $1$ and $2$.

• c) $\mathbb{R} \setminus \{-5\}$

  • This represents all real numbers except $-5$. On the number line, you would have a continuous line with an open circle at $-5$.

3. Represent each of the following sets on a number line:

• a) $[-2, 4] \cap (5, 7]$

  • The intersection of the interval $[-2, 4]$ and $(5, 7]$ results in an empty set, since these intervals do not overlap. The number line would not contain any shading or intervals.

• b) $[2, 7] \cap (-\infty, 5]$

  • The intersection of $[2, 7]$ and $(-\infty, 5]$ is $[2, 5]$, as it includes all points from 2 to 5. On the number line, this would be a shaded interval from 2 to 5, inclusive of both 2 and 5.

• c) $(-\infty, 2) \cap (-3, \infty)$

  • The intersection of $(-\infty, 2)$ and $(-3, \infty)$ is the interval $(-3, 2)$, which includes all points between $-3$ and $2$ (with $-3$ not included but 2 included). The number line would show this interval as a shaded region from just after $-3$ to $2$.

• d) $(-1, 4] \cup [3, 4)$

  • The union of $(-1, 4]$ and $[3, 4)$ is the interval $(-1, 4]$, as the part $[3, 4)$ is already included within the interval $(-1, 4]$. On the number line, this would be a shaded interval from just after $-1$ to $4$, inclusive of 4.

Would you like a more detailed explanation for any part of this? Or would you like to go through similar examples? Here are some questions to expand your understanding:

1. How do you handle intersections or unions when working with infinite sets?
2. What are the differences between open, closed, and half-open intervals on a number line?
3. How would you represent the set $\mathbb{R} \setminus \{n\}$ for any integer $n$?
4. What happens when you take the union of disjoint intervals?
5. How do you graph more complex set operations involving both union and intersection?

Tip: When drawing intervals on a number line, always clearly mark which endpoints are included (using closed or open circles) and ensure you identify the direction of any infinite bounds.