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Chapter 2: Problem 71

In Exercises 69-82, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. \(\\{x \mid x \in \mathbf{N}\) and \(30

Short Answer

Expert verified
The statement is true.

Step by step solution

01

Identify the elements of each set

The first set is: \(\{x \mid x \in \mathbf{N}\) and \(30<x<50\}\) and the second set is: \(\{x \mid x \in \mathbf{N}\) and \(30 \leq x \leq 50\}\). This means that the first set includes natural numbers between 30 and 50 (exclusive) and the second set includes natural numbers between 30 and 50 (inclusive).
02

List numbers in the set

The elements of the first set are: \(31, 32, 33, ..., 49\) and the elements of the second set are: \(30, 31, 32, ..., 49, 50\). As the first set does not include figures 30 and 50, which are part of the second set, the first set is a subset of the second set.
03

Apply the definition of a subset

A set A is a subset of set B if every element of A is also an element of B. From the previous step, we can observe that every element of the first set is also in the second set. Thus, the first set is a subset of the second set.

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Key Concepts

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

Subsets
In set theory, the concept of subsets is fundamental. Imagine you have a big box, which we call a set, filled with different elements or just distinct objects. A subset is a smaller box you can snipe from the big box, containing some or possibly all of the elements from the original box.

- **Definition**: A set A is considered a subset of a set B (written as \( A \subseteq B \)) if every element in A is also in B. - **Example**: For example, if A is the set \( \{1, 2, 3\} \) and B is the set \( \{1, 2, 3, 4, 5\} \), then A is clearly a subset of B because all elements of A appear in B.Subsets are a straightforward yet crucial concept for solving many mathematical problems. They help organize and compare different collections of objects, ensuring you understand how they relate to each other.
Natural Numbers
Natural numbers are the simplest numbers in mathematics. They begin from 1 and go upwards indefinitely: 1, 2, 3, and so on.

- **Understanding Natural Numbers**: These numbers do not include negative numbers, fractions, or decimals. They are often the first set of numbers that kids learn when they start counting. Natural numbers are denoted by the symbol \( \mathbf{N} \). - **Properties**: Natural numbers are closed under addition and multiplication, meaning you can add or multiply any of them without leaving the set of natural numbers. However, subtraction and division of natural numbers do not always yield a natural number. Knowing about natural numbers and their properties allows us to explore different topics in mathematics, including arithmetic and number theory.
Set Notation
Set notation is a standard way of expressing collections of objects. It is a crucial tool for clearly conveying which objects belong to a particular set and how sets relate to each other.

- **Basic Notation**: Sets are usually represented by curly braces. For instance, the set containing the numbers 1, 2, and 3 is written as \( \{1, 2, 3\} \).- **Set Builder Notation**: This is a compact way of expressing sets that release a rule or property that its members must follow. An example in our exercise is \( \{x \mid x \in \mathbf{N} \text{ and } 30 < x < 50\} \), which describes a set of natural numbers between 30 and 50.Using set notation, we can express the properties of sets and operations on them more precisely, aiding in the rigorous study and understanding of mathematics.

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Most popular questions from this chapter

A poll asks respondents the following question: Do you agree or disagree with this statement: In order to address the trend in diminishing male enrollment, colleges should begin special efforts to recruit men? a. Construct a Venn diagram with three circles that allows the respondents to be identified by gender (man or woman), education level (college or no college), and whether or not they agreed with the statement. b. Write the letter b in every region of the diagram that represents men with a college education who agreed with the statement. c. Write the letter c in every region of the diagram that represents women who disagreed with the statement. d. Write the letter \(\mathrm{d}\) in every region of the diagram that represents women without a college education who agreed with the statement. 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.

Describe the Venn diagram for two disjoint sets. How does this diagram illustrate that the sets have no common elements?

In Exercises 13-24, 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. \(C^{\prime} \cap\left(A \cup B^{\prime}\right)\)

In Exercises 1-12, 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. \((B \cup C)^{\prime} \cap A\)

This group activity is intended to provide practice in the use of Venn diagrams to sort responses to a survey. The group will determine the topic of the survey. Although you will not actually conduct the survey, it might be helpful to imagine carrying out the survey using the students on your campus. a. In your group, decide on a topic for the survey. b. Devise three questions that the pollster will ask to the people who are interviewed. c. Construct a Venn diagram that will assist the pollster in sorting the answers to the three questions. The Venn diagram should contain three intersecting circles within a universal set and eight regions. d. Describe what each of the regions in the Venn diagram represents in terms of the questions in your poll.
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