Problem 16 - Calculus with Concepts in Calculus

Question

Write the union of the two intervals as a single interval. $$ (-\infty, 0] \text { and }[0,3) $$

Step-by-Step Solution

Answer

The union of the intervals is \((-\infty, 3)\).

Step 1: Identify the Type of Intervals

The first interval is \((-\infty, 0]\), which is open on the left and closed on the right. The second interval is \([0, 3)\), which is closed on the left and open on the right.

Step 2: Analyze the Overlapping Point

Both intervals include the point 0. The first interval ends with 0 included, \(0]\), and the second interval starts with 0 included, \([0\). Thus, the point 0 is covered by both intervals.

Step 3: Determine the Complete Coverage

The first interval extends from \(-\infty\) to 0, and the second interval starts from 0 and goes to 3. Together, these intervals cover from \(-\infty\) to 3 without any gaps.

Step 4: Write the Single Union Interval

Since the intervals overlap or connect at point 0, combine them as \((-\infty, 3)\). The full range is open at \(-\infty\) and terminates just before 3.

Key Concepts

Interval Union

When dealing with interval notation in mathematics, an **interval union** is used to combine overlapping or connected intervals into a single, comprehensive interval. This is especially relevant when intervals have a common point or overlap completely or partially. To unite these intervals:

• Identify their common boundary points.
• Consider the openness or closedness of each endpoint.
• Combine them into a unified interval that reflects the full span covered by all individual intervals together.

For example, if you have intervals \((-\infty, 0]\) and \([0, 3)\), they overlap at 0. The union of these intervals would be \((-\infty, 3)\), showcasing how different intervals can seamlessly connect to cover a broader part of the real number line without gaps.

Open and Closed Intervals

Understanding the difference between **open and closed intervals** is fundamental in interval notation. An interval is essentially a portion of numbers along the real number line enclosed between two endpoints.

• **Open Intervals**: These do not include their endpoint(s). For instance, in the interval \((a, b)\), neither \(a\) nor \(b\) is included. The notation uses round brackets.
• **Closed Intervals**: These include their endpoint(s). For example, in the interval \([a, b]\), both \(a\) and \(b\) are part of the set. Square brackets denote this inclusion.
• **Half-Open/Half-Closed Intervals**: These contain one endpoint but not the other, like \([a, b)\) or \((a, b]\).

These distinctions play an important role in determining how intervals unite when they overlap. For example, in the intervals \([-\infty, 0]\) and \([0, 3)\), the point 0 is inclusive for both, aiding in the unification process.

Real Number Line

The **real number line** is a straight, infinitely extending line that visually represents all possible real numbers. Each point on this line corresponds to a real number, and it is used to illustrate various mathematical concepts like intervals, functions, and inequalities. Key aspects include:

• **Infinite Extent**: It stretches infinitely in both the positive and negative directions, often noted as \(\infty\) and \(-\infty\).
• **Visual Representation**: On a graph, intervals on the real number line are depicted by shaded regions with endpoints marked by open or closed circles, indicating whether these points are included in the interval.
• **Utility in Understanding**: Applying intervals to the real number line helps in visually comprehending concepts like union, intersection, and completeness of the number set.

Intervals such as \((-\infty, 3)\) illustrate the range of numbers included, starting from a point of indefinite negativity up to just shy of 3. By covering \( -\infty \) and continuing through every real number up to just before 3, these intervals help graphically demonstrate the potential range of values involved.