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Please do part C and D and please show step by step and explain

Please do part C and D and please show step by step and explain

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Please do part C and D and please show step by step and explain

**Exercise 15.2.22.** *Show that it is impossible to complete the following Cayley tables to make a group.* Four incomplete Cayley tables are presented. Each table is labeled (a), (b), (c), and (d), and features a 4x4 grid representing a binary operation on the set \(\{a, b, c, d\}\). Each table has some predefined entries in a lower triangular pattern, while other entries are left blank. **Table (a):** - The top row and first column are labeled \(a, b, c, d\). - Entries: - \(o(a, a) = a\) - \(o(b, b) = b\) - \(o(c, c) = c\) **Table (b):** - The top row and first column are labeled \(a, b, c, d\). - Entries: - \(o(a, a) = a\) - \(o(b, b) = b\) **Table (c):** - The top row and first column are labeled \(a, b, c, d\). - Entries: - \(o(b, c) = c\) - \(o(c, b) = b\) **Table (d):** - The top row and first column are labeled \(a, b, c, d\). - Entries: - \(o(a, b) = b\) - \(o(b, c) = c\) - \(o(c, d) = d\) In each table, the challenge is to fill in the blanks in a manner that satisfies the group axioms: closure, associativity, identity element, and inverse element for each member in the set. The exercise asks to show that this is not possible for any of these cases.
Transcribed Image Text:**Exercise 15.2.22.** *Show that it is impossible to complete the following Cayley tables to make a group.* Four incomplete Cayley tables are presented. Each table is labeled (a), (b), (c), and (d), and features a 4x4 grid representing a binary operation on the set \(\{a, b, c, d\}\). Each table has some predefined entries in a lower triangular pattern, while other entries are left blank. **Table (a):** - The top row and first column are labeled \(a, b, c, d\). - Entries: - \(o(a, a) = a\) - \(o(b, b) = b\) - \(o(c, c) = c\) **Table (b):** - The top row and first column are labeled \(a, b, c, d\). - Entries: - \(o(a, a) = a\) - \(o(b, b) = b\) **Table (c):** - The top row and first column are labeled \(a, b, c, d\). - Entries: - \(o(b, c) = c\) - \(o(c, b) = b\) **Table (d):** - The top row and first column are labeled \(a, b, c, d\). - Entries: - \(o(a, b) = b\) - \(o(b, c) = c\) - \(o(c, d) = d\) In each table, the challenge is to fill in the blanks in a manner that satisfies the group axioms: closure, associativity, identity element, and inverse element for each member in the set. The exercise asks to show that this is not possible for any of these cases.
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