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4. Prove for A = {x: x = 12k, k = Z}, B = {x : x = 3k, k = Z}, and C = {x: x = 4k, k = Z} that A BOC.

4. Prove for A = {x: x = 12k, k = Z}, B = {x : x = 3k, k = Z}, and C = {x: x = 4k, k = Z} that A BOC.

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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**Problem Statement:** 4. Prove for \( A = \{x : x = 12k, \, k \in \mathbb{Z}\} \), \( B = \{x : x = 3k, \, k \in \mathbb{Z}\} \), and \( C = \{x : x = 4k, \, k \in \mathbb{Z}\} \) that \( A = B \cap C \). **Discussion:** This problem asks us to demonstrate that the set \( A \) of integers that are multiples of 12 is equal to the intersection of the set \( B \) of multiples of 3 and the set \( C \) of multiples of 4. The solution involves showing that \( A \) includes all elements that are common in both \( B \) and \( C \), hence verifying the equality.
Transcribed Image Text:**Problem Statement:** 4. Prove for \( A = \{x : x = 12k, \, k \in \mathbb{Z}\} \), \( B = \{x : x = 3k, \, k \in \mathbb{Z}\} \), and \( C = \{x : x = 4k, \, k \in \mathbb{Z}\} \) that \( A = B \cap C \). **Discussion:** This problem asks us to demonstrate that the set \( A \) of integers that are multiples of 12 is equal to the intersection of the set \( B \) of multiples of 3 and the set \( C \) of multiples of 4. The solution involves showing that \( A \) includes all elements that are common in both \( B \) and \( C \), hence verifying the equality.
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