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.