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Chapter 0: Problem 3

List the elements in each set. $$ \\{x \mid x \text { is a natural number less than } 6\\} $$

Short Answer

Expert verified
1, 2, 3, 4, 5

Step by step solution

01

Understand the Set Notation

The set \( \{x \, | \, x \text{ is a natural number less than } 6\} \) means we need to find all natural numbers that are less than 6.
02

Identify Natural Numbers

Natural numbers are the positive integers starting from 1. So, the natural numbers are 1, 2, 3, 4, 5, 6, and so on.
03

List Natural Numbers Less than 6

Now, we list the natural numbers that are less than 6: 1, 2, 3, 4, and 5.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Set Notation
Set notation is a way to describe a collection of things, usually numbers, in a clear and concise manner. Sets are often denoted using curly braces, like this: \(...\). Inside the braces, you describe the elements of the set.
For instance, if you have a set containing all natural numbers less than 6, you can write it as \(\{x \, | \, x \text{ is a natural number less than } 6\}\). The vertical bar \(|\) means 'such that'.
This notation tells you to include all values 'x' that meet the condition after the bar. It's a powerful way to express a lot of information in a simple form.
  • Example:\(\{ x \, | \, x \text{ is a prime number less than 10}\}\)
  • Elements: 2, 3, 5, 7.
Natural Numbers
Natural numbers are a set of positive integers starting from 1 and going upwards: 1, 2, 3, 4, 5, and so on.
They are the numbers we naturally use for counting. Natural numbers do not include zero, negative numbers, fractions, or decimals.
Understanding what constitutes a natural number is crucial when working with sets that involve conditions related to these numbers.
  • Example of natural numbers less than 6: 1, 2, 3, 4, 5.
Mathematical Sets
A mathematical set is a well-defined collection of distinct objects, considered as an object in its own right. Sets can include numbers, people, letters, or any other items.
When working with numbers, sets are often used to group numbers with specific properties. For instance, the set of all even numbers or the set of all prime numbers.
Sets are usually written with curly braces, and the items within the set, called elements, are separated by commas.
  • Example: \(\{1, 2, 3, 4, 5\}\) - This set includes the first five natural numbers.
  • A set can also be described using a condition: \(\{x \, | \, x \text{ is a natural number less than 6}\}\)
Inequalities
Inequalities are mathematical expressions showing the relationship between two values where one value is less than, greater than, or not equal to another value.
They are critical when describing sets with conditions that restrict the values you can include. For example, to describe numbers less than a specific value, you use the '<' sign.
In our exercise, the inequality 'less than 6' tells you to include all numbers less than 6.
Here is how you can view inequalities:
  • <: Less than
  • 鈮�: Less than or equal to
  • >: Greater than
  • 鈮�: Greater than or equal to
Understanding inequalities helps you better grasp which numbers belong to a given set condition.

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Most popular questions from this chapter

Complete the table of fraction, decimal, and percent equivalents. $$\begin{array}{|c|c|c|}\hline \begin{array}{c}\text { Fraction in Lowest Terms } \\\\\text { (or Whole Number) }\end{array} & \text { Decimal } & \text { Percent } \\\\\hline \frac{1}{10}& & \\\\\hline\end{array}$$

Simplify each expression. $$ -3(5)^{2}-(-2)(-8) $$

Simplify each inequality if needed. Then determine whether the statement is true or false. $$ -6<7+3 $$

Find each square root. If it is not a real number, say so. $$ -\sqrt{0.49} $$

Simplify each expression. $$ |-6-5|(-8)-3^{2} $$
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