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Chapter 2: Problem 33

In Exercises 33-46, determine which sets are the empty set. \(\\{\varnothing, 0\\}\)

Short Answer

Expert verified
The set \( \{\varnothing, 0\} \) is not an empty set as it contains two elements: the empty set \(\varnothing\) and the number zero.

Step by step solution

01

Identify the elements in the given set.

The given set is \( \{\varnothing, 0\} \). It has two elements: \(\varnothing\) and 0.
02

Compare with the definition of an empty set.

By definition, an empty set or null set is a set that contains no elements. Looking at our given set, it is clear that this set contains two elements, hence it is not an empty set.

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Key Concepts

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

Null Set
The null set, also referred to as the empty set, is a fundamental concept in set theory. It is defined as a set that contains no elements at all. In mathematical notation, the null set is typically represented by the symbol \(\varnothing\) or by empty braces \(\{\}\). This set is unique in that it serves as a subset of every conceivable set. For example, even though the null set has nothing inside it, it is considered a part of any and all sets because it does not contain any elements to conflict with any given set's elements.
Set Theory
Set theory is the mathematical study of sets, which are collections of objects. This area of mathematics is crucial because it forms the foundation for more advanced concepts and is used extensively in nearly all areas of mathematics. A set is defined by its elements, which can be numbers, objects, or even other sets. Sets are usually denoted by brackets, such as \(\{1, 2, 3\}\), which represents a set containing the numbers 1, 2, and 3. In set theory, basic operations such as union, intersection, and difference help us understand relationships between different sets.
Set Elements
Elements within a set are the objects or numbers that make up the set. These elements can be anything: numbers, characters, or even other sets. Each element in a set is unique; duplicates are not considered. For example, the set \(\{a, b, c\}\) contains three distinct elements. There's an important distinction to be made: the notation \(\{\varnothing, 0\}\) represents a set with two elements, one of which is the null set and the other is the number zero. This demonstrates how different types of objects can coexist within a single set. Understanding what constitutes elements in a set is crucial for recognizing whether a set is empty.
Mathematical Definitions
Mathematical definitions provide a precise and unambiguous description of terms and concepts within mathematics. When dealing with set theory, definitions are vital for clarifying what is meant by concepts like the null set. A definition must be concise and clear to prevent misinterpretation. In set theory, the definition of a null set is critical because it establishes that the null set is one without any elements. Definitions in mathematics help structure our understanding and enable us to communicate complex ideas with simplicity and accuracy.

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Most popular questions from this chapter

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. \(A^{\prime} \cup C^{\prime}\)

Construct a Venn diagram illustrating the given sets. \(\begin{aligned} A &=\left\\{x_{3}, x_{9}\right\\} \\ B &=\left\\{x_{1}, x_{2}, x_{3}, x_{5}, x_{6}\right\\} \\ C &=\left\\{x_{3}, x_{4}, x_{5}, x_{6}, x_{9}\right\\} \\ U &=\left\\{x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}, x_{8}, x_{9}\right\\} \end{aligned}\)

Let $$ \begin{aligned} U &=\\{x \mid x \in \mathbf{N} \text { and } x<9\\} \\ A &=\\{x \mid x \text { is an odd natural number and } x<9\\} \\ B &=\\{x \mid x \text { is an even natural number and } x<9\\} \\ C &=\\{x \mid x \in \mathbf{N} \text { and } 1

A pollster conducting a telephone poll of a city's residents asked two questions: 1\. Do you currently smoke cigarettes? 2\. Regardless of your answer to question 1, would you support a ban on smoking in all city parks? a. Construct a Venn diagram that allows the respondents to the poll to be identified by whether or not they smoke cigarettes and whether or not they support the ban. b. Write the letter b in every region of the diagram that represents smokers polled who support the ban. c. Write the letter \(\mathrm{c}\) in every region of the diagram that represents nonsmokers polled who support the ban. d. Write the letter d in every region of the diagram that represents nonsmokers polled who do not support the ban.

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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