Consider the set L:= {(a,b) E Z²| (a – b) is a multiple of 3}. A proposed recursive definition for L is outlined below and the set it creates is called L'. Base Cases: {(-1,1), (0,0), (1,1)}cĽ' Recursive Rules: If (a,b) E L', then (a+3,b) E L' and (a,b– 3) e L' If this recursive definition is correct, then L' mistake, then prove that L' =L. Otherwise, edit the definition of L' and then prove that L' = L. L, but maybe a mistake was made. If there is no
Consider the set L:= {(a,b) E Z²| (a – b) is a multiple of 3}. A proposed recursive definition for L is outlined below and the set it creates is called L'. Base Cases: {(-1,1), (0,0), (1,1)}cĽ' Recursive Rules: If (a,b) E L', then (a+3,b) E L' and (a,b– 3) e L' If this recursive definition is correct, then L' mistake, then prove that L' =L. Otherwise, edit the definition of L' and then prove that L' = L. L, but maybe a mistake was made. If there is no
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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This is a discrete math problem. Please explain each step clearly, no cursive writing.
![Consider the set \( L := \{ (a, b) \in \mathbb{Z}^2 \mid (a-b) \text{ is a multiple of 3} \} \). A proposed recursive definition for \( L \) is outlined below, and the set it creates is called \( L' \).
**Base Cases:**
\[
\{ (-1, 1), (0, 0), (1, 1) \} \subset L'
\]
**Recursive Rules:**
If \( (a, b) \in L' \), then
\[
(a+3, b) \in L' \quad \text{and} \quad (a, b-3) \in L'
\]
If this recursive definition is correct, then \( L' = L \), but maybe a mistake was made. If there is no mistake, then prove that \( L' = L \). Otherwise, edit the definition of \( L' \) and then prove that \( L' = L \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F06a737e7-f57c-449b-94dc-ce666780f911%2F9f8002cb-3e9b-43a2-a4f7-54faeb5e2966%2Fhz8gho_processed.png&w=3840&q=75)
Transcribed Image Text:Consider the set \( L := \{ (a, b) \in \mathbb{Z}^2 \mid (a-b) \text{ is a multiple of 3} \} \). A proposed recursive definition for \( L \) is outlined below, and the set it creates is called \( L' \).
**Base Cases:**
\[
\{ (-1, 1), (0, 0), (1, 1) \} \subset L'
\]
**Recursive Rules:**
If \( (a, b) \in L' \), then
\[
(a+3, b) \in L' \quad \text{and} \quad (a, b-3) \in L'
\]
If this recursive definition is correct, then \( L' = L \), but maybe a mistake was made. If there is no mistake, then prove that \( L' = L \). Otherwise, edit the definition of \( L' \) and then prove that \( L' = L \).
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