Let 

*Vi* = *x*  **R** − 

1

*i*

 ≤ *x* ≤ 

1

*i*

 = − 

1

*i*

, 

1

*i*

 for each positive integer *i*. Find each of the following. (Enter your answers using interval notation.)

(a)

∪4*i* = 1*Vi* = 
 
(b)

∩4*i* = 1*Vi* = 
 

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

---

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.

Let Vi = x ∈ R : -1/i ≤ x ≤ 1/i for each positive integer i. Find each of the following. (a) ∪4i=1Vi = (b) ∩4i=1Vi = (Enter your answers using interval notation.)

Explore how to find the union and intersection of the intervals Vi = x ∈ R: -1/i ≤ x ≤ 1/i for i=1 to 4. Learn set theory concepts and interval notation with step-by-step solutions, including union and intersection operations. Suitable for high school students.

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