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Chapter 2: Problem 16

Express each set using the roster method. The set of months of the year that have exactly 30 days

Short Answer

Expert verified
\{'April', 'June', 'September', 'November'\}

Step by step solution

01

Understand the Problem

The set in question is 'The set of months of the year that have exactly 30 days'. From this, it is understood that it's necessary to identify which months in a year have exactly 30 days. Which are recognized as: April, June, September, November.
02

Apply the Roster Method

In the roster method, a set is expressed by listing elements between braces or curly brackets. Considering the previous step, the listed months: April, June, September, November are placed within these brackets. Thus forming the set.

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Key Concepts

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

Set Theory
Set theory is a fundamental concept in mathematics that deals with the study of collections of objects, known as sets. These objects can be anything, like numbers, letters, or even other sets. A set is usually represented by listing its elements inside curly brackets, such as \( \{1, 2, 3\} \). One key idea in set theory is that a set is defined entirely by its elements, meaning the order or repetition of elements doesn't matter. For example, \( \{1, 2, 3\} \) is the same as \( \{3, 2, 1\} \).

Some important properties to remember about sets include:
  • Well-defined collection: Every object is either a member of the set or not.
  • Equality: Two sets are equal if they contain precisely the same elements.
  • Subset: Set A is a subset of set B if all elements of A are also elements of B.
  • Union and Intersection: Operations that combine sets to form new sets.
Understanding these basics helps in identifying and working with sets in various mathematical contexts.
Roster Method
The roster method is a straightforward way of representing sets in set theory. Here, all the elements of the set are listed explicitly, one after the other, and enclosed within curly brackets. For instance, to represent a set containing the numbers 2, 4, and 6, we write it as \( \{2, 4, 6\} \). This method is also called the "list notation" because it literally lists out each element.

Some useful points about the roster method include:
  • Directness: It provides a clear and complete view of what belongs to the set.
  • Simplicity: Particularly effective for small sets with distinct elements.
  • Limitations: Not ideal for large or infinite sets due to impracticality of listing every element.
For the exercise at hand, the set of months with 30 days, expressed using the roster method, is \( \{\text{April}, \text{June}, \text{September}, \text{November}\} \), clearly showing each member of the set.
Months of the Year
The months of the year are a perfect real-world example for studying sets. There are 12 months in a year, each with a specific number of days. Understanding the differences among them helps in identifying particular subsets, like those with a specific number of days.
In a standard calendar year:
  • 31 days: January, March, May, July, August, October, December.
  • 30 days: April, June, September, November.
  • 28 days: February (except 29 in a leap year).
By examining the characteristics of each set of months, students can apply concepts like the roster method effectively, illustrating how theoretical methods connect to everyday phenomena.

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Most popular questions from this chapter

Let $$ \begin{aligned} U &=\\{\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e}, \mathrm{f}, \mathrm{g}, \mathrm{h}\\} \\ A &=\\{\mathrm{a}, \mathrm{g}, \mathrm{h}\\} \\ B &=\\{\mathrm{b}, \mathrm{g}, \mathrm{h}\\} \\ C &=\\{\mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e}, \mathrm{f}\\} \end{aligned} $$ Find each of the following sets. \((A \cup B) \cap(A \cup C)\)

A pollster conducting a telephone poll asked three questions: 1\. Are you religious? 2\. Have you spent time with a person during his or her last days of a terminal illness? 3\. Should assisted suicide be an option for terminally ill people? a. Construct a Venn diagram with three circles that can assist the pollster in tabulating the responses to the three questions. b. Write the letter b in every region of the diagram that represents all religious persons polled who are not in favor of assisted suicide for the terminally ill. c. Write the letter \(\mathrm{c}\) in every region of the diagram that represents the people polled who do not consider themselves religious, who have not spent time with a terminally ill person during his or her last days, and who are in favor of assisted suicide for the terminally ill. d. Write the letter \(d\) in every region of the diagram that represents the people polled who consider themselves religious, who have not spent time with a terminally ill person during his or her last days, and who are not in favor of assisted suicide for the terminally ill. e. Write the letter \(e\) in a region of the Venn diagram other than those in parts (b)-(d) and then describe who in the poll is represented by this region.

A pollster conducting a telephone poll at a college campus asked students two questions: 1\. Do you binge drink three or more times per month? 2\. Regardless of your answer to question 1, are you frequently behind in your school work? a. Construct a Venn diagram that allows the respondents to the poll to be identified by whether or not they binge drink and whether or not they frequently fall behind in school work. b. Write the letter \(b\) in every region of the diagram that represents binge drinkers who are frequently behind in school work. c. Write the letter c in every region of the diagram that represents students polled who do not binge drink but who are frequently behind in school work. d. Write the letter d in every region of the diagram that represents students polled who do not binge drink and who do not frequently fall behind in their school work.

Let $$ \begin{aligned} U &=\\{1,2,3,4,5,6,7\\} \\ A &=\\{1,3,5,7\\} \\ B &=\\{1,2,3\\} \\ C &=\\{2,3,4,5,6\\} \end{aligned} $$ Find each of the following sets. \(A \cap U\)

In Exercises 41-66, let $$ \begin{aligned} U &=\\{\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e}, \mathrm{f}, \mathrm{g}, \mathrm{h}\\} \\ A &=\\{\mathrm{a}, \mathrm{g}, \mathrm{h}\\} \\ B &=\\{\mathrm{b}, \mathrm{g}, \mathrm{h}\\} \\ C &=\\{\mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e}, \mathrm{f}\\} \end{aligned} $$ Find each of the following sets. \(A^{\prime} \cap B^{\prime}\)
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