For each i ∈ N+, let Ai = {−i, −i + 1, · · · , −1, 0, 1, · · · , i − 1, i}. What are U∞i=1 Ai and ∩∞i=1 Ai?

Let \( A_i \) be the set defined for each positive integer \( i \) as:

\[
A_i = \{-i, -i + 1, \dots, -1, 0, 1, \dots, i - 1, i\}
\]

We are tasked with finding:

1. \( \bigcup_{i=1}^{\infty} A_i \) (the union of all sets \( A_i \) for \( i \in \mathbb{N^+} \))
2. \( \bigcap_{i=1}^{\infty} A_i \) (the intersection of all sets \( A_i \) for \( i \in \mathbb{N^+} \))

### 1. Union: \( \bigcup_{i=1}^{\infty} A_i \)
Each \( A_i \) contains integers from \( -i \) to \( i \), so as \( i \) increases, the range of integers covered by \( A_i \) also increases.

- \( A_1 = \{-1, 0, 1\} \)
- \( A_2 = \{-2, -1, 0, 1, 2\} \)
- \( A_3 = \{-3, -2, -1, 0, 1, 2, 3\} \)
- \( \dots \)

As \( i \to \infty \), the union will include all integers, both positive and negative, since each \( A_i \) adds more integers to the set. Hence:

\[
\bigcup_{i=1}^{\infty} A_i = \mathbb{Z}
\]

### 2. Intersection: \( \bigcap_{i=1}^{\infty} A_i \)
The set \( A_i \) contains integers from \( -i \) to \( i \). As \( i \) increases, the range of integers grows, but every \( A_i \) contains the number 0, and as \( i \to \infty \), the only element that remains in all sets is 0. This is because, for any \( n \neq 0 \), there exists some \( i \) such that \( n \notin A_i \).

Thus:

\[
\bigcap_{i=1}^{\infty} A_i = \{0\}
\]

### Final Answers:
\[
\bigcup_{i=1}^{\infty} A_i = \mathbb{Z}
\]
\[
\bigcap_{i=1}^{\infty} A_i = \{0\}
\]

Would you like more details or have any questions about this?

Here are 5 related questions to expand on this concept:
1. What happens if \( A_i \) is defined as \( A_i = \{i, i+1, \dots, 2i\} \)?
2. How does the union and intersection change if the sets are finite instead of infinite?
3. What is the union and intersection of sets \( B_i = \{2^i, 2^i+1, \dots, 2^{i+1}-1\} \)?
4. Can you describe the union and intersection of an arbitrary sequence of sets where each \( A_i \) is bounded?
5. How does the behavior of unions and intersections change in continuous sets?

**Tip**: The union of an increasing sequence of sets typically captures all elements in their eventual range, while intersections tend to get smaller, focusing on common elements.

Explore the union and intersection of infinite sets where Ai is defined as {−i, −i + 1, ..., 0, ..., i}. Learn how the union results in all integers, while the intersection results in the set containing only 0. Suitable for students at the undergraduate level or advanced high school learners.

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