Problem 18 - Calculus with Concepts in Calculus

Question

Write the union of the two intervals as a single interval. $$ \left(-\infty, \frac{1}{2}\right] \text { and }(0, \infty) $$

Step-by-Step Solution

Answer

The union is \((-\infty, \infty)\).

Step 1: Identify Intervals

The intervals given are \((-\infty, \frac{1}{2}]\) and \((0, \infty)\). These describe all numbers less than or equal to \(\frac{1}{2}\) and all numbers greater than 0.

Step 2: Analyze Overlapping Area

Both intervals cover some common ground after 0. The interval \((0, \frac{1}{2}]\) exists in both intervals.

Step 3: Determine Start of Union

The starting point of the union is the smaller of the two intervals, which is unbounded on the left, \((-\infty, \frac{1}{2}]\).

Step 4: Determine End of Union

The ending point of the union is \((0, \infty)\), which is unbounded on the right. Since both intervals overlap after 0, the union is unbounded to infinity.

Step 5: Write the Union

Combine the intervals to form the union, which is \((-\infty, \infty)\) because every real number is included in either or both intervals.

Key Concepts

Real Numbers

Real numbers are the building blocks of our number system. They include all the numbers that you are used to working with, such as whole numbers, fractions, and irrational numbers. The real number system can be divided into several subsets. These subsets include:

• Natural Numbers: 1, 2, 3, ...
• Whole Numbers: 0, 1, 2, 3, ...
• Integers: ..., -2, -1, 0, 1, 2, ...
• Rational Numbers: Numbers that can be expressed as fractions, like \( \frac{1}{2}, \frac{-4}{5}, 2 \)
• Irrational Numbers: Numbers that cannot be written as a simple fraction, such as \( \sqrt{2}, \pi \)

Real numbers can be positive, negative, or zero and are usually represented on a number line. They span all possible values from negative infinity to positive infinity. That's why you'll often see expressions like \( (-\infty, c) \) or \( (d, \infty) \) used to describe intervals or sets of real numbers.

Inequalities

Inequalities are mathematical expressions used to show the relative size or order of two values. These expressions come with four main inequality symbols:

• \( < \): Less Than
• \( > \): Greater Than
• \( \leq \): Less Than or Equal To
• \( \geq \): Greater Than or Equal To

Understanding inequalities is crucial when dealing with intervals. For example, the notation \( (-\infty, \frac{1}{2}] \) denotes all real numbers less than or equal to \( \frac{1}{2} \). The notation \( (0, \infty) \) represents all real numbers greater than 0.
In the context of real numbers, these inequalities help define the \'range\' of values that are included within a particular interval. They allow us to understand the extent to which our intervals stretch on the real number line. Recognizing overlapping and unbounded intervals is essential for solving problems related to interval unions.

Set Notation

Set notation is a systematic way to express collections of objects, usually numbers. When you're dealing with intervals on the real number line, set notation becomes handy to succinctly describe them.
An interval like \( (-\infty, \frac{1}{2}] \) can be thought of as a set of all numbers "belonging" to the interval \( x \leq \frac{1}{2} \). Meanwhile, \( (0, \infty) \) is the set of all real numbers \( x > 0 \).
When you unite or "join" these two sets, you use the \'union\' operator, denoted by the symbol \( \cup \). The union encompasses all elements from each set, without duplication. For the given intervals, the union results in \( (-\infty, \infty) \), which includes every real number, demonstrating the power and simplicity of set notation in expressing such comprehensive unions.