The following is written in my mathematical proofs book "Z4 consists of the four classes (sets of integers) [0],[1],[2],[3], where for r ∈ {0, 1, 2, 3}, [r] = {3q + r : q ∈ Z}" Im confused on why it's 3q+r and not 4q+r, I know there is a general rule but i cannot find it, I know why it consists of 4 classes, but somehow i dont get the "{3q + r : q ∈ Z}" part, please explain step by step.
The following is written in my mathematical proofs book "Z4 consists of the four classes (sets of integers) [0],[1],[2],[3], where for r ∈ {0, 1, 2, 3}, [r] = {3q + r : q ∈ Z}" Im confused on why it's 3q+r and not 4q+r, I know there is a general rule but i cannot find it, I know why it consists of 4 classes, but somehow i dont get the "{3q + r : q ∈ Z}" part, please explain step by step.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.5: Graphs Of Functions
Problem 56E
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The following is written in my mathematical proofs book
"Z4 consists of the four classes (sets of integers)
[0],[1],[2],[3], where for r ∈ {0, 1, 2, 3}, [r] = {3q + r : q ∈ Z}"
Im confused on why it's 3q+r and not 4q+r, I know there is a general rule but i cannot find it, I know why it consists of 4 classes, but somehow i dont get the "{3q + r : q ∈ Z}" part, please explain step by step.
![Z4 of integers modulo 4. Recall that Z4 consists of the four classes (sets of integers)
[0], [1], [2], [3], where for r = {0, 1, 2, 3}, [r] = {3q + r : q € Z}. Furthermore, if a, b = [r]
for some r with 0 ≤ r ≤ 3, then [a] = [b] and a = b (mod 4).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8e237d3f-b8e6-4775-a6f9-5671b153aef2%2F4a6b0ed2-93d7-4401-9867-945d53225358%2Fisd65e8_processed.png&w=3840&q=75)
Transcribed Image Text:Z4 of integers modulo 4. Recall that Z4 consists of the four classes (sets of integers)
[0], [1], [2], [3], where for r = {0, 1, 2, 3}, [r] = {3q + r : q € Z}. Furthermore, if a, b = [r]
for some r with 0 ≤ r ≤ 3, then [a] = [b] and a = b (mod 4).
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