602 CHAPTER 17 EQUIVALENCE RELATIONS AND EQUIVALENCE CLASSES (a) Let B = {1, 2, 3, 4, 5} and S = {(1, 1), (1,4), (2, 2), (2,3), (3,2), (3, 3), (4,1), (4,4), (5,5)}. Assume (without proof) that S is an equivalence relation on B. Find the equivalence class of each element of B. (b) Let C = {1, 2, 3, 4, 5} and define ~ by x~Cyx+y is even. Assume (without proof) that ~ is an equivalence relation on C. Find the equivalence class of each element of C. (c) Draw the arrow diagrams for the relations in R in Example 17.4.5, and for the relations in parts (a) and (b) of this exercise.
602 CHAPTER 17 EQUIVALENCE RELATIONS AND EQUIVALENCE CLASSES (a) Let B = {1, 2, 3, 4, 5} and S = {(1, 1), (1,4), (2, 2), (2,3), (3,2), (3, 3), (4,1), (4,4), (5,5)}. Assume (without proof) that S is an equivalence relation on B. Find the equivalence class of each element of B. (b) Let C = {1, 2, 3, 4, 5} and define ~ by x~Cyx+y is even. Assume (without proof) that ~ is an equivalence relation on C. Find the equivalence class of each element of C. (c) Draw the arrow diagrams for the relations in R in Example 17.4.5, and for the relations in parts (a) and (b) of this exercise.
Advanced Engineering Mathematics
10th Edition
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Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Please do part A,B,C and please show step by step and explain
![**Example 17.4.5.** Suppose \( A = \{1, 2, 3, 4, 5\} \) and
\[
R = \{(1, 1), (1, 3), (1, 4), (2, 2), (2, 5), (3, 1), (3, 3), (3, 4), (4, 1), (4, 3), (4, 4), (5, 2), (5, 5)\}
\]
One can verify that \( R \) is an equivalence relation on \( A \). The equivalence classes are:
\[
[1] = [3] = [4] = \{1, 3, 4\}, \quad [2] = [5] = \{2, 5\}.
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F892e817a-9b32-4eeb-b8fc-5dd7ffde6479%2Fa023e8e6-79ab-4788-af9f-06dc731bc32a%2Fxqxlwjw_processed.png&w=3840&q=75)
Transcribed Image Text:**Example 17.4.5.** Suppose \( A = \{1, 2, 3, 4, 5\} \) and
\[
R = \{(1, 1), (1, 3), (1, 4), (2, 2), (2, 5), (3, 1), (3, 3), (3, 4), (4, 1), (4, 3), (4, 4), (5, 2), (5, 5)\}
\]
One can verify that \( R \) is an equivalence relation on \( A \). The equivalence classes are:
\[
[1] = [3] = [4] = \{1, 3, 4\}, \quad [2] = [5] = \{2, 5\}.
\]
![**Equivalence Relations and Equivalence Classes**
(a) Let \( B = \{1, 2, 3, 4, 5\} \) and
\[ S = \{(1, 1), (1, 4), (2, 2), (2, 3), (3, 2), (3, 3), (4, 1), (4, 4), (5, 5)\} .\]
Assume (without proof) that \( S \) is an equivalence relation on \( B \). Find the equivalence class of each element of \( B \).
(b) Let \( C = \{1, 2, 3, 4, 5\} \) and define \(\sim\) by
\[ x \sim y \iff x + y \text{ is even}. \]
Assume (without proof) that \(\sim\) is an equivalence relation on \( C \). Find the equivalence class of each element of \( C \).
(c) Draw the arrow diagrams for the relations in \( R \) in Example 17.4.5, and for the relations in parts (a) and (b) of this exercise.
**Diagram Explanation:**
For parts (a) and (b), the user is asked to create arrow diagrams representing the equivalence relations \( S \) and \(\sim\), respectively.
- **Part (a) Diagram**: Illustrate the set \( B \) with arrows connecting each element within its equivalence class according to the relation \( S \).
- **Part (b) Diagram**: Illustrate the set \( C \) and show which elements are connected by the equivalence relation \(\sim\) so that the sum of the connected elements is even.
These diagrams help to visually represent the relationships defined by the equivalence relations in the given sets.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F892e817a-9b32-4eeb-b8fc-5dd7ffde6479%2Fa023e8e6-79ab-4788-af9f-06dc731bc32a%2F69r5wel_processed.png&w=3840&q=75)
Transcribed Image Text:**Equivalence Relations and Equivalence Classes**
(a) Let \( B = \{1, 2, 3, 4, 5\} \) and
\[ S = \{(1, 1), (1, 4), (2, 2), (2, 3), (3, 2), (3, 3), (4, 1), (4, 4), (5, 5)\} .\]
Assume (without proof) that \( S \) is an equivalence relation on \( B \). Find the equivalence class of each element of \( B \).
(b) Let \( C = \{1, 2, 3, 4, 5\} \) and define \(\sim\) by
\[ x \sim y \iff x + y \text{ is even}. \]
Assume (without proof) that \(\sim\) is an equivalence relation on \( C \). Find the equivalence class of each element of \( C \).
(c) Draw the arrow diagrams for the relations in \( R \) in Example 17.4.5, and for the relations in parts (a) and (b) of this exercise.
**Diagram Explanation:**
For parts (a) and (b), the user is asked to create arrow diagrams representing the equivalence relations \( S \) and \(\sim\), respectively.
- **Part (a) Diagram**: Illustrate the set \( B \) with arrows connecting each element within its equivalence class according to the relation \( S \).
- **Part (b) Diagram**: Illustrate the set \( C \) and show which elements are connected by the equivalence relation \(\sim\) so that the sum of the connected elements is even.
These diagrams help to visually represent the relationships defined by the equivalence relations in the given sets.
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