Exercise 17.5.15. Find x, y € Z12 such that a 0 and y = 0, but x - y = 0.

Elements Of Modern Algebra
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ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
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Chapter2: The Integers
Section2.2: Mathematical Induction
Problem 10E
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Please do Exercise 17.5.15 and please show step by step and explain

Proposition 17.5.12. For any n € N+, we have
Zn = {0, 1, 2,..., n
and 0, 1, 2,.. n I are all distinct.
Exercise 17.5.13. Prove Proposition 17.5.12. It is sufficient to show (a)
0,1,2,..., n − 1 are distinct; and (b) for any integer, the equivalence class
k is one of 0, 1, 2,...,n - 1.
Exercise 17.5.14. Using the definitions of addition, subtraction, and mul-
tiplication given in part (c) of Definition 17.5.11, make tables that show the
results of:
(a) addition modulo 4.
(b) subtraction modulo 5.
(c) multiplication modulo 6.
Exercise 17.5.15. Find r, y € Z12 such that z ‡0 and y ‡ 0, but x· y =
= 0.
Transcribed Image Text:Proposition 17.5.12. For any n € N+, we have Zn = {0, 1, 2,..., n and 0, 1, 2,.. n I are all distinct. Exercise 17.5.13. Prove Proposition 17.5.12. It is sufficient to show (a) 0,1,2,..., n − 1 are distinct; and (b) for any integer, the equivalence class k is one of 0, 1, 2,...,n - 1. Exercise 17.5.14. Using the definitions of addition, subtraction, and mul- tiplication given in part (c) of Definition 17.5.11, make tables that show the results of: (a) addition modulo 4. (b) subtraction modulo 5. (c) multiplication modulo 6. Exercise 17.5.15. Find r, y € Z12 such that z ‡0 and y ‡ 0, but x· y = = 0.
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