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Consider the sets: A = {m € Z/m = 6k+9 for some k € Z} B = {m € Z/m = 3w for some w € Z} Is AC B? Is B C A? Is A = B? Justify your answer below using a proof.

Consider the sets: A = {m € Z/m = 6k+9 for some k € Z} B = {m € Z/m = 3w for some w € Z} Is AC B? Is B C A? Is A = B? Justify your answer below using a proof.

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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5. Consider the sets: \[ A = \{ m \in \mathbb{Z} \mid m = 6k + 9 \text{ for some } k \in \mathbb{Z} \} \] \[ B = \{ m \in \mathbb{Z} \mid m = 3w \text{ for some } w \in \mathbb{Z} \} \] Is \( A \subseteq B \)? Is \( B \subseteq A \)? Is \( A = B \)? Justify your answer below using a proof.
Transcribed Image Text:5. Consider the sets: \[ A = \{ m \in \mathbb{Z} \mid m = 6k + 9 \text{ for some } k \in \mathbb{Z} \} \] \[ B = \{ m \in \mathbb{Z} \mid m = 3w \text{ for some } w \in \mathbb{Z} \} \] Is \( A \subseteq B \)? Is \( B \subseteq A \)? Is \( A = B \)? Justify your answer below using a proof.
Expert Solution
Step 1

Given- A=m∈Z|m=6k+9 for some k∈Z and B=m∈Z|m=3w for some w∈Z.

Let x∈A

Therefore, x=6k+9 for some k∈Z.

x=32k+3 for some k∈Z.

x=3w1 where w1=2k+3 which is of the form 3w.

Therefore, x∈B.

Hence, 

A⊆B.

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