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We are given the following sets in the problem:

- \( A_r = \{(x, y) \in \mathbb{R} \times \mathbb{R} \mid x^2 + y^2 = r^2\} \)
- \( B_r = \{(x, y) \in \mathbb{R} \times \mathbb{R} \mid x^2 + y^2 < r^2\} \)
- \( C_r = \{(x, y) \in \mathbb{R} \times \mathbb{R} \mid x^2 + y^2 > r^2\} \)

These define three sets related to circles in the plane:
- \( A_r \) is the set of points on the circle of radius \( r \) centered at the origin.
- \( B_r \) is the set of points inside the circle of radius \( r \).
- \( C_r \) is the set of points outside the circle of radius \( r \).

The goal of part (a) is to determine the union and intersection of \( A_r \) over all \( r \in I \), where \( I = (0, \infty) \). 

### 1. Union of \( A_r \) for \( r \in I \):
The union \( \bigcup_{r \in I} A_r \) represents all points \( (x, y) \) that lie on some circle with radius \( r \in (0, \infty) \). This means we are covering all points at any finite distance from the origin.

Since every point on each circle corresponds to some radius \( r \), the union of all circles is the entire plane minus the origin, because no circle contains the origin itself.

Thus, 
\[
\bigcup_{r \in I} A_r = \mathbb{R}^2 \setminus \{(0,0)\}.
\]

### 2. Intersection of \( A_r \) for \( r \in I \):
The intersection \( \bigcap_{r \in I} A_r \) would contain only those points that are on every circle of radius \( r \). However, there is no single point that lies on every circle, as the radius is different for each circle.

Therefore, the intersection is empty:
\[
\bigcap_{r \in I} A_r = \emptyset.
\]

### Final Answer:
- \( \bigcup_{r \in I} A_r = \mathbb{R}^2 \setminus \{(0, 0)\} \)
- \( \bigcap_{r \in I} A_r = \emptyset \)

Would you like a further breakdown of any part? Here are 5 related questions to expand on this:

1. What would be the union and intersection of \( B_r \) for \( r \in I \)?
2. How does this problem change if the interval \( I \) is closed, i.e., \( I = [0, \infty) \)?
3. Can you interpret these sets geometrically in terms of distance from the origin?
4. What happens to the union and intersection if we define \( A_r \) with inequalities like \( B_r \) or \( C_r \)?
5. How would these sets change if we used a different norm (like the maximum norm) instead of the Euclidean distance?

**Tip:** Understanding how unions and intersections behave with geometric shapes like circles can help in visualizing complex sets.

Let I denote the interval (0, ∞). For each r ∈ I, define the sets A_r, B_r, and C_r as: A_r = {(x, y) ∈ R × R | x^2 + y^2 = r^2}, B_r = {(x, y) ∈ R × R | x^2 + y^2 < r^2}, C_r = {(x, y) ∈ R × R | x^2 + y^2 > r^2}. Determine ∪_{r ∈ I} A_r and ∩_{r ∈ I} A_r.

Explore how to determine the union and intersection of sets representing circles in the plane. Understand the geometric properties of circles defined by x^2 + y^2 = r^2, and how these sets combine for various radii.

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