https://file.aiquickdraw.com/mathbot/file/1725978979409ba0b34fe.jpg

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.

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### 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 \( -2 < x < 2 \).

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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.

Graphically represent the sets A, B, and C in the plane, and represent A ∩ B and the complement R^2 \ (B ∪ C) in the plane.

This problem involves representing sets A, B, and C graphically in the Cartesian plane. Set A is a parabola, B and C are regions defined by inequalities. The task also includes finding the intersection A ∩ B and the complement R^2 \ (B ∪ C).

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