Problem 33 - Intermediate Algebra

Question

List the elements of each set. For example, the elements of \(\\{x \mid x\) is a natural number less than 4\(\\}\) can be listed as \(\\{1,2,3\\}\). \(\\{x \mid x\) is a natural number less than 3\(\\}\)

Step-by-Step Solution

Answer

\(\{1, 2\}\)

Step 1: Understanding Natural Numbers

Natural numbers are counting numbers starting from 1. They are 1, 2, 3, 4, and so on. Unlike whole numbers, natural numbers do not include 0.

Step 2: Interpreting the Condition

The condition provided is that the number, denoted as \(x\), must be less than 3. This means we are looking for natural numbers that are smaller than 3.

Step 3: Listing Elements

To list the numbers that satisfy the condition, start from the smallest natural number and include numbers in increasing order until the condition is no longer satisfied. In this case, these numbers are 1 and 2.

Step 4: Forming the Set

Collect all the elements that satisfy the condition into a set notation. The numbers 1 and 2 are the elements. Thus, the set is \(\{1, 2\}\).

Key Concepts

Set Notation

Set notation is a powerful way to group numbers or objects that share a common property. In mathematics, a set is a collection of elements, typically enclosed within curly braces \( \{ \} \). Set notation often includes a description of the property that the elements of the set satisfy. This description is written within the set notation and separated by a vertical bar "\(|\)," which can be read as "such that." For example, the set \( \{x \mid x\) is a natural number less than 3\} contains all numbers \(x\) that are natural numbers and smaller than 3. The vertical bar separates the variable \(x\) from its defining condition, making set notation a concise way to express complex ideas about numbers and their properties. It is essential to understand the notation used, as it provides clear guidance on which elements should be included in the set.

Listing Elements

After understanding the condition defined in the set notation, the next step is to list the elements that meet this condition. Think of listing elements as finding all numbers or objects that fit the criteria specified.
For our example, we want to identify the natural numbers less than 3. Natural numbers begin at 1 and do not include 0. Hence, we start counting from 1.

• Start with 1: As 1 is the first natural number, it's included in the list.


• Then 2: The next number, 2, is also less than 3, so it fits the condition.


• Stop before 3: Since 3 is not less than 3, it cannot be included.

Thus, the elements of the set \( \{x \mid x \text{ is a natural number less than 3}\} \) are simply \( \{1, 2\} \). Remembering to systematically list numbers until the given condition is no longer true ensures accuracy.

Number Conditions

Understanding number conditions is crucial when working with set notation. A number condition specifies what numbers are eligible to be in a set.
In our exercise, the condition was that \(x\) must be a natural number less than 3.

• Identifying the condition: Recognize the numerical limit, which in this case is "less than 3."


• Determine natural numbers: Realize that this involves numbers like 1, 2, 3, and so on, but excludes non-positive integers.


• Apply the condition: Only numbers 1 and 2 meet the criteria of being both natural numbers and less than 3. Although 3 is a natural number, it does not meet the criterion "less than 3."

By understanding the conditions set forth, one can correctly identify which numbers belong and avoid mistakes in listing members of the set. This skill becomes even more valuable when dealing with more complex conditions in mathematics.