/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 9 For each of the following sets, ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 9

For each of the following sets, determine whether 2 is an element of that set. $$ \begin{array}{l}{\text { a) }\\{x \in \mathbf{R} | x \text { is an integer greater than } 1\\}} \\ {\text { b) }\\{x \in \mathbf{R} | x \text { is the square of an integer }\\}} \\ {\text { c) }\\{2,\\{2\\}\\}} \\ {\text { e) }\\{\\{2\\},\\{2,\\{2\\}\\}\\}} & {\text { f) }\\{\\{2\\},\\{\\{2\\}\\}\\}}\end{array} $$

Short Answer

Expert verified
2 is an element of sets (a) and (c).

Step by step solution

01

- Analyze Set (a)

Set (a) is defined as \({ x \in \mathbf{R} | x \text{ is an integer greater than }1 \}\). An integer greater than 1 includes 2, 3, 4, etc. Hence, 2 is an element of this set.
02

- Analyze Set (b)

Set (b) is defined as \({ x \in \mathbf{R} | x \text{ is the square of an integer }\}\). The squares of integers are 0, 1, 4, 9, 16, etc. Since 2 is not a perfect square, 2 is not an element of this set.
03

- Analyze Set (c)

Set (c) is \({ 2, \{2\} \}\). This set contains the number 2 as one of its elements. Therefore, 2 is an element of this set.
04

- Analyze Set (e)

Set (e) is \({ \{2\}, \{2, \{2\}\} \}\). This set contains \{2\} and \{2, \{2\}\} as its elements. Since these are subsets and not the number 2 itself, 2 is not an element of this set.
05

- Analyze Set (f)

Set (f) is \({ \{2\}, \{\{2\}\} \}\). It contains \{2\} and \{\{2\}\} as its elements. Since these are subsets and not the number 2 itself, 2 is not an element of this set.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Integer
An integer is any whole number, including negative numbers, zero, and positive numbers. When we talk about integers, we're referring to numbers like -3, 0, 1, 2, and so on. Integers do not include fractions or decimals. In mathematical problems, integers are often represented by the symbol \(\textbf{Z}\). Remember, integers are a key building block in math for counting, ordering, and performing various arithmetic operations. When we say a number is greater than 1, it can be 2, 3, 4, and so forth. Here, 2 is indeed an integer greater than 1, fulfilling the criteria in Set (a).
Subset
A subset is a set where all elements are also found in another set. It's a smaller part of a larger set. The notation \(\textbf{A} \subseteq \textbf{B}\) indicates that set A is a subset of set B. Let's use Set (e) as an example. It includes \(\text \textbraceleft 2 \textbraceright, \textbraceleft 2, \text \textbraceleft 2 \textbraceright \textbraceright\). This means we have sets within sets. Even though \{2\} is a subset inside the set, the number 2 itself is not directly an element. Getting comfortable with subsets can make it much easier to understand nesting and how elements relate to each other.
Real Numbers
Real numbers include all the numbers on the number line. This encompasses whole numbers, fractions, and irrational numbers. Real numbers are represented by the symbol \(\textbf{R}\). Some examples include -2, 0.5, 3.14, and 2. Real numbers can be positive, negative, or zero. They are crucial in both pure and applied math due to their comprehensive nature. For instance, in all sets listed, \(\textbf{R}\) signifies that the numbers in question are real numbers, whether they are integers or other forms. Understanding real numbers helps in understanding the broader scope of number sets and their properties.
Perfect Square
A perfect square is an integer that can be expressed as the square of another integer. For example, 9 is a perfect square because it can be written as 3x3. If you take any integer n and square it (multiply it by itself), the result is a perfect square: \({n}^2\). Some perfect squares are 0, 1, 4, 9, 16, and so on. From our exercise, Set (b) asks if 2 can be a perfect square. Since there is no integer n such that \({n}^2 = 2\), 2 is not a perfect square, hence it's not an element of that set. Recognizing perfect squares involves understanding both multiplication and the properties of integers.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Data are transmitted over a particular Ethernet network in blocks of 1500 octets (blocks of 8 bits). How many blocks are required to transmit the following amounts of data over this Ethernet network? (Note that a byte is a synonym for an octet, a kilobyte is 1000 bytes, and a megabyte is \(1,000,000\) bytes.) a) 150 kilobytes of data b) 384 kilobytes of data c) 1.544 megabytes of data d) 45.3 megabytes of data

Find the domain and range of these functions. a) the function that assigns to each pair of positive integers the maximum of these two integers b) the function that assigns to each positive integer the number of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 that do not appear as decimal digits of the integer c) the function that assigns to a bit string the number of times the block 11 appears d) the function that assigns to a bit string the numerical position of the first 1 in the string and that assigns the value 0 to a bit string consisting of all 0s

What is the cardinality of each of these sets? $$ \begin{array}{ll}{\text { a) } \emptyset} & {\text { b) }\\{\emptyset\\}} \\\ {\text { c) }\\{\emptyset,\\{\emptyset\\}\\}} & {\text { d) }\\{\emptyset,\\{\emptyset\\},\\{\emptyset,\\{\emptyset\\}\\}\\}}\end{array} $$

List the first 10 terms of each of these sequences. a) the sequence obtained by starting with 10 and obtaining each term by subtracting 3 from the previous term b) the sequence whose \(n\) th term is the sum of the first \(n\) positive integers c) the sequence whose \(n\) th term is \(3^{n}-2^{n}\) d) the sequence whose \(n\) th term is \(\lfloor\sqrt{n}\rfloor\) e) the sequence whose first two terms are 1 and 5 and each succeeding term is the sum of the two previous terms f) the sequence whose \(n\) th term is the largest integer whose binary expansion (defined in Section 4.2) has \(n\) bits (Write your answer in decimal notation.) g) the sequence whose terms are constructed sequentially as follows: start with 1 , then add \(1,\) then multiply by \(1,\) then add \(2,\) then multiply by \(2,\) and so on h) the sequence whose \(n\) th term is the largest integer \(k\) such that \(k ! \leq n\)

List the first 10 terms of each of these sequences. a) the sequence that begins with 2 and in which each successive term is 3 more than the preceding term b) the sequence that lists each positive integer three times, in increasing order c) the sequence that lists the odd positive integers in in- creasing order, listing each odd integer twice d) the sequence whose nth term is \(n !-2^{n}\) e) the sequence that begins with 3, where each succeeding term is twice the preceding term f ) the sequence whose first term is 2, second term is 4, and each succeeding term is the sum of the two preceding terms g) the sequence whose nth term is the number of bits in the binary expansion of the number n (defined in Section 4.2) h) the sequence where the nth term is the number of letters in the English word for the index n
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Statistics

Read Explanation

Applied Mathematics

Read Explanation

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.