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Chapter 1: Problem 96

Translate to an algebraic expression. $$ \\{x | x=2 n, n \text { is an integer }\\} $$

Short Answer

Expert verified
The algebraic expression is \(2n\).

Step by step solution

01

Identify the Set Description

The given set \(\{x | x = 2n, n \text{ is an integer }\}\) describes all numbers \(x\) that can be written as \(2n\), where \(n\) is any integer.
02

Understand the Role of \(n\)

Here, \(n\) is a variable representing any integer. This means \(n\) can take any integer value (e.g., -2, -1, 0, 1, 2, etc.).
03

Formulate the Algebraic Expression

Since \(x\) is defined as \(2n\), and \(n\) can be any integer, the algebraic expression for the set is simply \(2n\). This indicates a general even number.

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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 mathematical language used to describe and represent sets, which are collections of objects. In this exercise, the set is given by \( \{x | x=2 n, n \text { is an integer }\} \). This notation tells us that we are looking at a set of elements, specifically all numbers that can be expressed in the form of \(2n\), where \(n\) is an integer.

Here's the breakdown:
  • The curly braces \{ ... \} denote a set.
  • The symbol \textbar \ (read as 'such that') separates the elements from the condition they must satisfy.
  • The expression \( x = 2n \) describes the form of the elements in the set.
  • The phrase 'n is an integer' indicates the range of values that \( n \) can take.
Using set notation provides a clear, concise way to represent collections of numbers or objects that follow a specific rule.
Integers
Integers are a fundamental concept in math, representing whole numbers that can be positive, negative, or zero. They are denoted by \( \mathbb{Z} \). In this exercise, the variable \( n \) represents any integer, which means it can take on values like:

  • -3, -2, -1
  • 0
  • 1, 2, 3
Understanding that \( n \) can be any integer helps us realize that \( x \) can be any even number, as the expression \( x = 2n \) covers all possible values obtained by multiplying 2 with an integer.
Even Numbers
Even numbers are integers that are divisible by 2. In mathematical terms, an even number can be expressed as \( 2n \), where \( n \) is any integer.

Some key points about even numbers are:
  • They include both positive and negative integers, as well as zero (e.g., -4, -2, 0, 2, 4).
  • Every even number has a form where it can be expressed by multiplying 2 with another integer.
In the exercise, the set \( \{ x | x = 2n , n \text { is an integer }\} \) represents all these even numbers. Here, \( x = 2n \) tells us that any number \( x \) in this set must be an even number because it is obtained by multiplying an integer \( n \) by 2. This set is essentially the set of all even integers.

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