Problem 5 - A Gentle Introduction to the Art of Mathematics

Question

For each positive integer \(n,\) we'll define an interval \(I_{n}\) by $$I_{n}=[-n, 1 / n)$$ Find the union and intersection of all the intervals in this infinite family. $$\begin{array}{l} \bigcup_{n \in \mathbb{N}} I_{n}= \\ \bigcap_{n \in \mathbb{N}} I_{n}= \end{array}$$

Step-by-Step Solution

Answer

The union is all real numbers less than 0. The intersection is the empty set.

Step 1: Understanding the Interval

For each positive integer n, the interval is defined as \[ I_{n} = [-n, \frac{1}{n}) \]which means it includes all the numbers from -n to just before \( \frac{1}{n} \) including -n and excluding \( \frac{1}{n} \).

Step 2: Analyze the Union of All Intervals

To find the union of all intervals, we need to determine all the values that are covered by any of the intervals. As n increases, the left endpoint (-n) becomes more negative, and the right endpoint (\frac{1}{n}) becomes smaller (approaches 0 from the positive side). Hence, the union of all intervals, \( \bigcup_{n \in \mathbb{N}} I_{n} \), contains all real numbers less than 0, and does not include 0.

Step 3: Analyze the Intersection of All Intervals

To find the intersection of all intervals, we need to find the values that are included in every interval. As n increases, the interval \( I_{n} \) becomes wider but the right endpoint remains less than \( \frac{1}{n} \), which becomes arbitrarily close to 0. Therefore, there is no single value that is in every interval because for any non-zero value, there exists an n such that \( x \) is not included in \( I_{n} \). The only number common in all intervals would be the empty set because no number is part of every interval.

Key Concepts

Union of Intervals

In mathematics, the union of intervals combines all the elements from multiple intervals into a single set. Think of union like this: it collects all the numbers that appear in any given interval.
With our given intervals, it means we take all values covered by any interval without repeating them.

Intersection of Intervals

The intersection of intervals, on the other hand, is all about what numbers are common among all given intervals. It's like finding the overlap in multiple sets.
For our intervals, since each interval changes with every new value of n, finding a common number is tricky. The only way a number can be in all intervals is if each interval includes that number, which doesn't happen here. As intervals expand, their overlapping part shrinks to nothing. Hence, the intersection of our intervals is an empty set.

Real Numbers

Real numbers include all the numbers we usually work with: positive, negative, and zero. They form an unbroken line on the number line, including every possible decimal or fraction.
In the exercise, we're working within this set of real numbers but focusing on specific ranges (intervals).
Understanding how these intervals behave within the realm of real numbers helps us learn how to manage and analyze continuous data.