Problem 34 - Algebra for College Students

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 greater than 3\(\\}\)

Step-by-Step Solution

Answer

The set is \( \{4, 5, 6, 7, \ldots\} \).

Step 1: Understand the Set Notation

The set is given in set-builder notation: \( \{x \mid x \text{ is a natural number greater than 3}\} \). This means we are looking for numbers that belong to the set of natural numbers and are greater than 3.

Step 2: Define Natural Numbers

Natural numbers are positive integers starting from 1, such as 1, 2, 3, 4, and so on. Therefore, we are considering natural numbers greater than 3.

Step 3: Identify Elements Greater Than 3

In the set \( \{x \mid x \text{ is a natural number greater than 3}\} \), we identify the natural numbers starting from the next integer after 3. These numbers include 4, 5, 6, 7, and so forth.

Step 4: List the Elements

Write down the natural numbers that meet the condition \( x > 3 \). These are 4, 5, 6, 7, and continue indefinitely. Therefore, the set can be listed as \( \{4, 5, 6, 7, \ldots\} \).

Key Concepts

Understanding Natural Numbers

Natural numbers are the basic counting numbers we use in everyday life. These are the numbers that you first learn when starting your arithmetic journey. They begin from 1 and go on in an endless sequence: 1, 2, 3, 4, 5, and so forth.
Natural numbers do not include zero or any negative numbers. They also notably exclude fractions and decimals. This sequence connects directly to counting, as you can look at objects and count them using these numbers.
When you hear the term 'greater than' with respect to natural numbers, it implies a number that is further along in this infinite sequence. So, if a number is said to be greater than 3, we include all numbers after 3 in our list, such as 4, 5, 6, etc.

Decoding Set-Builder Notation

Set-builder notation is a concise way to define sets by specifying a property that elements of the set have in common.
In the example given, the set is expressed as \( \{x \mid x \text{ is a natural number greater than 3} \} \). This tells us what the numbers in the set represent and what condition they need to satisfy.

• The \( \{x \mid \} \) structure indicates we are defining a set based on conditions.
• The condition here is "\( x \text{ is a natural number greater than 3} \)".

This means we're only interested in natural numbers that are more than 3. Set-builder notation is extremely useful in mathematics as it provides a clear, standard way to define large or infinite sets without listing all elements explicitly.

Listing Elements of a Set

To list elements of a set, you need to identify the values that satisfy the conditions provided in the set notation.
Once you know the description or pattern, you can express these elements precisely. In our example, the condition was that \( x \) is "a natural number greater than 3."

• Start listing natural numbers immediately after 3: 4, 5, 6, 7, etc.
• The ellipsis "\( \ldots \)" denotes that the pattern continues indefinitely.

This method implies a sequence that you understand will never end, which is how infinite sets are typically communicated. Listing elements helps in visualizing the set and understanding its scope, especially when working through complex problems.