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Chapter 0: Problem 10

List the elements in each set. $$ \\{k \mid k \text { is an odd integer less than } 1\\} $$

Short Answer

Expert verified
The set is \( \{-3, -1\} \).

Step by step solution

01

Understand the Set Notation

The notation \( \{k \mid k \text{ is an odd integer less than } 1\} \) describes a set of elements \( k \) such that \( k \) is an odd integer and \( k < 1 \).
02

Identify Odd Integers

Odd integers are numbers like -3, -1, 1, 3, etc. They are not divisible by 2.
03

Find Odd Integers Less Than 1

Now, filter the list of odd integers to find those that are less than 1. The odd integers less than 1 are ..., -3, -1.
04

List the Elements

Compile the identified odd integers less than 1 into a set: \( \{-3, -1\} \).

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Key Concepts

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

set notation
In mathematics, set notation is a system used to define a collection of objects or elements. Sets are often represented using curly braces {} and contain elements that meet specific criteria. For example, the set \(\{k \mid k \text{ is an odd integer less than } 1\}\) describes a set of elements k such that k is an odd integer and \(k < 1\). In this set, the vertical bar \(|\) can be read as 'such that,' indicating the conditions for membership in the set.

This standardized method helps in clearly communicating which elements are included in a set, making it easier to analyze and work with mathematical problems.
properties of integers
Integers are a fundamental part of mathematics and include all whole numbers, both positive and negative, as well as zero. They have several important properties:
  • Odd and Even Integers: Odd integers (e.g., -3, -1, 1, 3) are not divisible by 2, while even integers (e.g., -2, 0, 2, 4) are divisible by 2.
  • Addition and Subtraction: Adding or subtracting even integers always produces even results. Adding or subtracting an odd integer and an even integer results in an odd number. Adding or subtracting two odd integers results in an even number.
  • Multiplication: Multiplying two even integers or two odd integers results in an even number.
  • Order and Magnitude: Integers can be arranged in ascending or descending order, showcasing their mathematical relationships.
Fully understanding these properties allows you to easily manipulate and identify integers that meet specific conditions, such as finding all odd integers less than 1.
filtering integers
Filtering integers involves selecting specific integers from a larger set based on given criteria. For instance, to find the odd integers less than 1 from the set of all odd integers, follow these steps:
  • Identify the odd integers: ..., -3, -1, 1, 3,
  • Apply the condition: Only include those odd integers that are less than 1.
  • Compile the results: The integers that meet the criteria are -3 and -1.
This method of filtering helps in narrowing down larger sets to subsets that are of particular interest or importance. In the given exercise, the integers in the set \(\{k \mid k \text{ is an odd integer less than } 1\}\) are simply \(\{-3, -1\}.\)

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