Problem 11 - Precalculus

Question

Write the set using interval notation. $$ \\{x \mid x \neq 0,2\\} $$

Step-by-Step Solution

Answer

The set in interval notation is \((-\infty, 0) \cup (0, 2) \cup (2, \infty)\).

Step 1: Understand the Set Notation

The given set notation \( \{x \mid x eq 0,2\} \) describes all real numbers except for 0 and 2. We are tasked to represent this set using interval notation.

Step 2: Break Down the Set into Intervals

Since the set excludes 0 and 2, we split the real numbers into three distinct intervals that exclude these numbers: 1. From negative infinity to 0 (excluding 0) 2. From 0 to 2 (excluding 0 and 2) 3. From 2 to positive infinity (excluding 2).

Step 3: Write Each Interval in Interval Notation

For each part identified in Step 2, write in interval notation:- The interval from negative infinity to 0 is \((-\infty, 0)\).- The interval from 0 to 2 is \((0, 2)\).- The interval from 2 to infinity is \((2, \infty)\).

Step 4: Combine the Intervals in Interval Notation

Use the union symbol \(\cup\) to combine the intervals, as each section is separate from the others. The set in interval notation is: \((-\infty, 0) \cup (0, 2) \cup (2, \infty)\).

Key Concepts

Set Notation

Set notation is a method used to describe a group of elements, often numbers, that share a common property or characteristic. It's like a mathematical language used to specify a set clearly. This notation usually involves a pair of braces, \{\}, with elements or conditions listed between them.

For example, with the expression \( \{x \mid x eq 0, 2\} \), we're looking at all numbers "\(x\)" that don't equal 0 or 2. The vertical bar "\(\mid\)" reads as "such that," so the set reads as: "The set of all \(x\) such that \(x\) is not equal to 0 or 2."

Set notation is effective in providing a compact representation of vast number sets without specifying each number individually. It's particularly useful in mathematics when you're dealing with infinite sets or complex conditions.

Real Numbers

Real numbers are an incredibly broad category of numbers that include almost every kind of number you can think of. They're the numbers you encounter in everyday life, like fractions, whole numbers, and irrational numbers, such as \(\pi\).

To simplify, real numbers include:

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

Real numbers fill the line in a mathematical number line without any gaps. The beauty of real numbers is that they can go on forever in both the positive and negative direction, creating continuity in mathematics.

Intervals

Intervals define a range of real numbers between two endpoints. In interval notation, an interval is expressed using brackets and parentheses to depict inclusivity or exclusivity of the endpoints.

For the set expression \(\{x \mid x eq 0, 2\}\), we break it down into intervals, excluding 0 and 2. For example:

• \(( -\infty, 0 )\) indicates numbers from negative infinity up to, but not including, 0.
• \((0, 2)\) covers numbers between 0 and 2, excluding 0 and 2 themselves.
• \((2, \infty)\) signifies numbers from above 2 to positive infinity, excluding 2.

Using intervals ensures we accurately describe a continuous range of numbers, clarifying the parts of the real number line that meet certain criteria.

Union Symbol

The union symbol, written as \(\cup\), comes into play when you need to connect separate intervals or sets into a single, cohesive unit.

In our exercise, when transforming set notation into interval notation, you will notice multiple separate intervals resulted from excluding 0 and 2: \(( -\infty, 0 )\), \((0, 2)\), and \((2, \infty)\).

To combine these distinct parts into one expression, you use the union symbol. This process joins the separated sections into a complete description: \(( -\infty, 0 ) \cup (0, 2) \cup (2, \infty)\).

Using the union symbol simplifies and unifies the presentation of multiple intervals, allowing you to express them as a singular entity efficiently.