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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
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 \( L' = L \).
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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