/*! 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 3 Assume that the universal set is... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 3

Assume that the universal set is the set of integers. Let $$\begin{array}{ll} A=\\{-3,-2,2,3\\}, & B=\left\\{x \in \mathbb{Z} \mid x^{2} \leq 9\right\\} \\ C=\\{x \in \mathbb{Z} \mid x \geq-3\\}, & D=\\{1,2,3,4\\} \end{array}$$ In each of the following, fill in the blank with one or more of the symbols \(\subset\), \(\subseteq, \nsubseteq,=, \neq, \in,\) or \(\notin\) so that the resulting statement is true. For each blank, include all symbols that result in a true statement. If none of these symbols makes a true statement, write nothing in the blank.

Short Answer

Expert verified
In conclusion: 1. A \(鈯哱) B 2. D \(鈮燶) A 3. C \(鈯哱) B, C \(鈮燶) B 4. A \(鈯哱) C 5. 1 \(鈭圽) D

Step by step solution

01

Case 1: A ____ B

First, let's compare sets A and B: A = {-3, -2, 2, 3} B = {x 鈭� 鈩� : x虏 鈮� 9} => B = {-3, -2, -1, 0, 1, 2, 3} Clearly, every element of A is present in B. Therefore, A is a subset of B. The symbol we can use is "鈯�"
02

Case 2: D ____ A

Now, let's compare sets D and A: D = {1, 2, 3, 4} A = {-3, -2, 2, 3} In this case, some elements of D are in A and some are not. Therefore, we can say that A is not equal to D. The symbol we can use is "鈮�"
03

Case 3: C ____ B

Next, let's compare sets C and B: C = {x 鈭� 鈩� : x 鈮� -3} => C = {-3, -2, -1, 0, 1, 2, 3, 4, 5, 6,...} B = {-3, -2, -1, 0, 1, 2, 3} Here, B is a subset of C but not equal to it. The symbols we can use are "鈯�" and "鈮�"
04

Case 4: A ____ C

Now, let's compare sets A and C: A = {-3, -2, 2, 3} C = {-3, -2, -1, 0, 1, 2, 3, 4, 5, 6,...} In this case, A is a subset of C as all elements of A are found in C. The symbol we can use is "鈯�"
05

Case 5: 1 ____ D

Lastly, let's compare the element 1 and set D: 1 is an integer D = {1, 2, 3, 4} Since 1 is an element of set D, we can say that 1 is in D. The symbol we can use is "鈭�" In summary: 1. A 鈯� B 2. D 鈮� A 3. C 鈯� B, C 鈮� B 4. A 鈯� C 5. 1 鈭� D

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.

Subset and Superset
In set theory, understanding the relationship between two sets is crucial. A subset is a portion of a set which contains some or all elements of another set. If set A is a subset of set B, then every element in A is also in B. For instance, if we have two sets where A = {1, 2} and B = {1, 2, 3, 4}, then A is a subset of B, represented as A \(subseteq\) B.

Conversely, a superset contains all elements of a subset plus potentially more. Using the previous example, B is a superset of A. Every subset has at least one superset 鈥� the set itself 鈥� since sets are always considered to be subsets of themselves. This is denoted by <= in set notation. Understanding this concept helps in solving problems such as determining set relationships between A and B in our exercise, where clearly A was a subset of B.
Inequalities in Set Theory
Inequalities in set theory provide a way to compare the size of sets or to express the relationship between elements and sets. These relationships are symbolized by expressions like \(subset\), \subseteq, =, \eq, \in, and \otin. For example, when we say A \subseteq B, we express that set A is a subset of set B, which can be either proper (strict subset) or improper (equal to B). A proper subset, denoted by A \(subset\) B, means A contains fewer elements than B.

Moreover, if we determine that A does not equal B, as in the case of sets C and B from the exercise, we denote it as C \eq B. Similarly, we use \in to indicate that an element is in a set and \otin to indicate that it is not. These symbols help to frame precise statements within set theory, a foundational concept for solving many mathematical problems.
Integer Sets
Integer sets are collections of numbers that do not have fractional or decimal parts, including both positive and negative whole numbers along with zero. For example, the set D = {1, 2, 3, 4} in our original problem consists entirely of integers. The universal set of integers is denoted by \(\mathbb{Z}\).

In the context of our exercise, integer sets are compared to determine subset relations or to express the contained elements using inequalities, such as B = {x \in \mathbb{Z} \mid x^2 \leq 9}, which is the set of all integers whose square is less than or equal to nine. Working with integer sets often involves understanding number properties and operations, which is fundamental knowledge for any student of mathematics.

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

Let \(A, B,\) and \(C\) be subsets of some universal set \(U\) (a) Draw two general Venn diagrams for the sets \(A, B,\) and \(C .\) On one, shade the region that represents \(A-(B-C),\) and on the other, shade the region that represents \((A-B)-C .\) Based on the Venn diagrams, make a conjecture about the relationship between the sets \(A-(B-C)\) and \((A-B)-C .\) (Are the two sets equal? If not, is one of the sets a subset of the other set?) (b) Prove the conjecture from Exercise (7a).

For each positive real number \(r,\) define \(T_{r}\) to be the closed interval \(\left[-r^{2}, r^{2}\right]\) That is, \(T_{r}=\left\\{x \in \mathbb{R} \mid-r^{2} \leq x \leq r^{2}\right\\}\) Let \(\Lambda=\\{m \in \mathbb{N} \mid 1 \leq m \leq 10\\}\). Use either interval notation or set builder notation to specify each of the following sets: * (a) \(\bigcup_{k \in \Lambda} T_{k}\) (c) \(\bigcup_{r \in \mathbb{R}^{+}} T_{r}\) (e) \(\bigcup_{k \in \mathbb{N}} T_{k}\) *(b) \(\bigcap_{k \in \Lambda} T_{k}\) (d) \(\bigcap_{r \in \mathbb{R}^{+}} T_{r}\) (f) \(\bigcap_{k \in \mathbb{N}} T_{k}\)

Let \(C=\\{x \in \mathbb{Z} \mid x \equiv 7(\bmod 9)\\}\) and \(D=\\{x \in \mathbb{Z} \mid x \equiv 1(\bmod 3)\\}\). (a) List at least five different elements of the set \(C\) and at least five elements of the set \(D\). (b) Is \(C \subseteq D\) ? Justify your conclusion with a proof or a counterexample. (c) Is \(D \subseteq C\) ? Justify your conclusion with a proof or a counterexample.

To help with the proof by induction of Theorem \(5.5,\) we first prove the following lemma. (The idea for the proof of this lemma was illustrated with the discussion of power set after the definition on page \(222 .\) ) Lemma 5.6. Let \(A\) and \(B\) be subsets of some universal set. If \(A=B \cup\\{x\\},\) where \(x \notin B\), then any subset of \(A\) is either a subset of \(B\) or a set of the form \(C \cup\\{x\\},\) where \(C\) is a subset of \(B\) Proof. Let \(A\) and \(B\) be subsets of some universal set, and assume that \(A=\) \(B \cup\\{x\\}\) where \(x \notin B\). Let \(Y\) be a subset of \(A\). We need to show that \(Y\) is a subset of \(B\) or that \(Y=C \cup\\{x\\},\) where \(C\) is some subset of \(B\). There are two cases to consider: (1)\(x\) is not an element of \(Y,\) and (2)\(x\) is an element of \(Y\) Case 1: Assume that \(x \notin Y\). Let \(y \in Y .\) Then \(y \in A\) and \(y \neq x .\) Since $$A=B \cup\\{x\\}.$$ this means that \(y\) must be in \(B\). Therefore, \(Y \subseteq B\). Case 2: Assume that \(x \in Y .\) In this case, let \(C=Y-\\{x\\} .\) Then every element of \(C\) is an element of \(B\). Hence, we can conclude that \(C \subseteq B\) and that \(Y=C \cup\\{x\\}\) Cases (1) and (2) show that if \(Y \subseteq A,\) then \(Y \subseteq B\) or \(Y=C \cup\\{x\\},\) where \(C \subseteq B\). To begin the induction proof of Theorem \(5.5,\) for each nonnegative integer \(n,\) we let \(P(n)\) be, "If a finite set has exactly \(n\) elements, then that set has exactly \(2^{n}\) subsets." (a) Verify that \(P(0)\) is true. (This is the basis step for the induction proof.) (b) Verify that \(P(1)\) and \(P(2)\) are true. (c) Now assume that \(k\) is a nonnegative integer and assume that \(P(k)\) is true. That is, assume that if a set has \(k\) elements, then that set has \(2^{k}\) subsets. (This is the inductive assumption for the induction proof.) Let \(T\) be a subset of the universal set with card \((T)=k+1,\) and let \(x \in T .\) Then the set \(B=T-\\{x\\}\) has \(k\) elements. Now use the inductive assumption to determine how many subsets \(B\) has. Then use Lemma 5.6 to prove that \(T\) has twice as many subsets as \(B\). This should help complete the inductive step for the induction proof.

For each natural number \(n,\) let \(A_{n}=\\{k \in \mathbb{N} \mid k \geq n\\} .\) Assuming the universal set is \(\mathbb{N}\), use the roster method or set builder notation to specify each of the following sets: (a) \(\bigcap_{j=1}^{5} A_{j}\) (e) \(\bigcup_{j=1}^{5} A_{j}\) (b) \(\left(\bigcap_{j=1}^{5} A_{j}\right)^{c}\) $$ \text { (f) }\left(\bigcup_{j=1}^{5} A_{j}\right)^{c} $$ (c) \(\bigcap_{j=1}^{5} A_{j}^{c}\) (g) \(\bigcap_{j \in \mathbb{N}} A_{j}\) (d) \(\bigcup_{j=1}^{5} A_{j}^{c}\) (h) \(\bigcup_{j \in \mathbb{N}} A_{j}\)
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Logic and Functions

Read Explanation

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

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.