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6. Let n ≥ 2 be an integer. Two integers a, b € Z are congruent mod n and denoted as a = b (mod n), if n | (a − b). 6a. Show that this congruent relation is an equivalence relation on the set Z of integers.

6. Let n ≥ 2 be an integer. Two integers a, b € Z are congruent mod n and denoted as a = b (mod n), if n | (a − b). 6a. Show that this congruent relation is an equivalence relation on the set Z of integers.

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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6. Let n ≥ 2 be an integer. Two integers a, b € Z are congruent mod n and denoted as a = b (mod n), if n | (a − b). 6a. Show that this congruent relation is an equivalence relation on the set Z of integers. 6b. Denote by à := {b € Z | b = a (mod n)} the equivalence class containing a (surely, a € ā). This a is called a representative of the class ā. Show that a = a +nZ = {a+ns s € Z}. 6c. Define a + b := a + b, ā × b := ab. Show that the addition and multiplication above are well defined, i.e., show: if a' =ā and b' = 6 then a'+b' = a + b and a'b' = ab. 6d. Let Z/(n) = {ā|a € Z} be the set of equivalence classes. Show that (Z/(n), +, ×,0,1) is a ring.
Transcribed Image Text:6. Let n ≥ 2 be an integer. Two integers a, b € Z are congruent mod n and denoted as a = b (mod n), if n | (a − b). 6a. Show that this congruent relation is an equivalence relation on the set Z of integers. 6b. Denote by à := {b € Z | b = a (mod n)} the equivalence class containing a (surely, a € ā). This a is called a representative of the class ā. Show that a = a +nZ = {a+ns s € Z}. 6c. Define a + b := a + b, ā × b := ab. Show that the addition and multiplication above are well defined, i.e., show: if a' =ā and b' = 6 then a'+b' = a + b and a'b' = ab. 6d. Let Z/(n) = {ā|a € Z} be the set of equivalence classes. Show that (Z/(n), +, ×,0,1) is a ring.
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