Problem 12 - Precalculus

Question

Write the set using interval notation. $$ \\{x \mid x \neq 2,-2\\} $$

Step-by-Step Solution

Answer

The interval notation is \((-inity, -2) \cup (-2, 2) \cup (2, inity)\).

Step 1: Understand the Set

The set given is \{x \mid x eq 2, -2\}, which includes all real numbers except for \(x = 2\) and \(x = -2\). This means that every number except these two is part of the set.

Step 2: Break the Set into Intervals

To exclude the points \(x = 2\) and \(x = -2\), the set can be divided into three intervals: one before \(-2\), one between \(-2\) and \(2\), and one after \(2\).

Step 3: Write the Intervals in Notation

The intervals are written in interval notation by using parentheses to denote that endpoints \(-2\) and \(2\) are not included. Therefore, the three intervals are: 1. \((-inity, -2)\) 2. \((-2, 2)\) 3. \((2, inity)\)

Step 4: Combine the Intervals

Because the three intervals are separate and not connected, we use the union symbol (\(\cup\)) to combine them into one complete set. The interval notation for the set is: \((-inity, -2) \cup (-2, 2) \cup (2, inity)\).

Key Concepts

Real Numbers

Real numbers are a fundamental concept in mathematics. They include all the numbers we can think of that are on the number line. This includes whole numbers, fractions, decimals, and irrational numbers. Imagine every possible point you can mark on an endless line, and those represent real numbers.
Understanding real numbers is crucial because they form the basis for functions, calculus, and other advanced mathematical concepts. Real numbers are not limited to numbers that can be written as simple fractions or integers, but they also include irrational numbers such as \(\pi\) and \(\sqrt{2}\). These numbers cannot be perfectly expressed as a ratio of two whole numbers but are still part of the real number system.

• Whole numbers: 0, 1, 2, 3, ...
• Fractions: \(\frac{1}{2}\), \(\frac{3}{4}\), ...
• Decimals: 0.75, -2.5, ...
• Irrational numbers: \(\pi\), \(\sqrt{2}\), ...

When working with real numbers, it's important to understand that certain operations like division by zero don't apply. Additionally, real numbers on the number line extend infinitely in both directions, covering both negative and positive values.

Set Notation

Set notation is a mathematical language used to describe and work with sets. A set is essentially a collection of distinct objects grouped together, which can be anything from numbers to shapes or even concepts.
In set notation, a set is usually written with curly braces. For example, \[\{ a, b, c \}\] is a simple set containing elements \(a\), \(b\), and \(c\).
When we speak about the set of numbers from the exercise, \[\{x \mid x eq 2, -2 \}\], it describes all elements \(x\) such that \(x\) is not equal to \(2\) or \(-2\).

• Element: An object within a set.
• Membership: The objects or elements that are included in the set.
• Description: A condition or rule that elements of the set satisfy.

A deeper understanding of set notation allows you to comprehend different mathematical problems and analyze groups and collections of objects or numbers effectively.

Union of Intervals

The union of intervals is a way to combine separate intervals into a single set. In mathematics, the union of intervals is denoted by the symbol \(\cup\). This operation allows us to group together multiple segments of numbers on the number line that are not necessarily adjacent.
Using the exercise example, we have three distinct intervals: \((-\infty, -2)\), \((-2, 2)\), \( (2, \infty)\). Each of these represents a range of real numbers:

• \((-\infty, -2)\): All real numbers less than \(-2\).
• \((-2, 2)\): All real numbers between \(-2\) and \(2\).
• \((2, \infty)\): All real numbers greater than \(2\).

By taking the union of these intervals, \(\{x \mid x eq 2, -2\}\), we efficiently represent all real numbers except \(2\) and \(-2\). This highlights the flexibility of using union to piece together non-overlapping parts of a set into a comprehensive whole. When working in mathematics, the concept of union is vital for creating complex sets and understanding how different numeric ranges interact.