The image contains a mathematical problem with three sets $A$, $B$, and $C$ defined in $\mathbb{R}^2$ along with two tasks:

Problem: Given sets $A$, $B$, and $C$ are defined as follows:

• $A = \{(x, y) \in \mathbb{R}^2 : y = x^2 + 1\}$
• $B = \{(x, y) \in \mathbb{R}^2 : |x| \leq 2, |y + 1| > 1\}$
• $C = \{(x, y) \in \mathbb{R}^2 : |x| > 2\}$

You are asked to: a) Graphically represent the sets $A$, $B$, and $C$ in the plane. b) Graphically represent the set $A \cap B$ and the complement $\mathbb{R}^2 \setminus (B \cup C)$ in the plane.

Step-by-step breakdown:

1. Set $A$:

   • This is the graph of the function $y = x^2 + 1$, which is a parabola opening upwards with vertex at $(0, 1)$.

2. Set $B$:

   • The condition $|x| \leq 2$ defines a vertical strip between $x = -2$ and $x = 2$.
   • The condition $|y + 1| > 1$ can be rewritten as $y > 0$ or $y < -2$, excluding the horizontal band $-2 < y \leq 0$.

3. Set $C$:

   • This is the region where $|x| > 2$, meaning the area outside the vertical strip $-2 < x < 2$.

To represent the sets visually:

• $A$ is a parabola.
• $B$ is the region between $x = -2$ and $x = 2$, excluding $-2 < y \leq 0$.
• $C$ is the region outside the strip $-2 < x < 2$.

For part (b), you'll need to:

• Find the intersection $A \cap B$, which is the region where both the parabola and the constraints of $B$ hold.
• Represent $\mathbb{R}^2 \setminus (B \cup C)$, the complement of the union of $B$ and $C$, which will be the remaining area.

Would you like a graphical solution or further details on the steps? Here are some related questions:

1. How do you calculate intersections of curves and regions graphically?
2. How would you solve similar problems in three dimensions?
3. How does the complement of a union work mathematically?
4. What is the geometric interpretation of inequalities in two-dimensional space?
5. Can you use computational tools to graph these sets?

Tip: For graphical representation, understanding the boundaries and constraints of each set is crucial in determining where they overlap or remain separate.