Let's break down the problem step by step.

We are given:

$V_i = \left\{ x \in \mathbb{R} : -\frac{1}{i} \leq x \leq \frac{1}{i} \right\}$

for each positive integer $i$. This means $V_i$ represents the interval:

$V_i = \left[ -\frac{1}{i}, \frac{1}{i} \right]$

The task is to find the following:

(a) $\bigcup_{i=1}^{4} V_i$

This is the union of the sets $V_1$, $V_2$, $V_3$, and $V_4$. Let's first write out each set:

• $V_1 = \left[ -1, 1 \right]$
• $V_2 = \left[ -\frac{1}{2}, \frac{1}{2} \right]$
• $V_3 = \left[ -\frac{1}{3}, \frac{1}{3} \right]$
• $V_4 = \left[ -\frac{1}{4}, \frac{1}{4} \right]$

Since we are taking the union, we consider all the values that are in any of these intervals. The largest interval is $V_1 = \left[ -1, 1 \right]$, so the union of all these intervals is:

$\bigcup_{i=1}^{4} V_i = \left[ -1, 1 \right]$

(b) $\bigcap_{i=1}^{4} V_i$

This is the intersection of the sets $V_1$, $V_2$, $V_3$, and $V_4$. Let's consider the intersection, which is the set of values that belong to all the intervals.

The interval that is contained in all the others is the smallest interval, $V_4 = \left[ -\frac{1}{4}, \frac{1}{4} \right]$. So, the intersection of all these intervals is:

$\bigcap_{i=1}^{4} V_i = \left[ -\frac{1}{4}, \frac{1}{4} \right]$

Final Answers:

(a) $\bigcup_{i=1}^{4} V_i = \left[ -1, 1 \right]$

(b) $\bigcap_{i=1}^{4} V_i = \left[ -\frac{1}{4}, \frac{1}{4} \right]$

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Would you like more details on any part of the process? Here are some related questions for practice:

1. What would $\bigcup_{i=1}^{5} V_i$ be?
2. What would $\bigcap_{i=1}^{3} V_i$ be?
3. How does the union of intervals behave when adding more terms?
4. How does the intersection of intervals behave as $i$ increases?
5. What happens to the size of $V_i$ as $i$ increases?

Tip: In set theory, the union operation includes all elements in any of the sets, while the intersection only includes elements common to all sets.