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Let S-(xeZ vmod12-8) and T (xe2| 4x). Prove that SCT but TgS La s-(rez| 3reZz, x-12r+7) and T-(xeZ| 3re Z, x- 3-2) Prove that ScT, but T g S.

Let S-(xeZ vmod12-8) and T (xe2| 4x). Prove that SCT but TgS La s-(rez| 3reZz, x-12r+7) and T-(xeZ| 3re Z, x- 3-2) Prove that ScT, but T g S.

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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### Problems involving Subsets and Modular Arithmetic #### Problem 5 Let \( S = \{ x \in \mathbb{Z} \mid x \mod 12 = 8 \} \) and \( T = \{ x \in \mathbb{Z} \mid 4 \mid x \} \). **Task**: Prove that \( S \subseteq T \), but \( T \not\subseteq S \). #### Solution Steps: 1. **Understanding \( S \)**: - The set \( S \) consists of all integers \( x \) that, when divided by 12, leave a remainder of 8. 2. **Understanding \( T \)**: - The set \( T \) consists of all integers \( x \) that are divisible by 4. 3. **Proving \( S \subseteq T \)**: - Every element \( x \in S \) can be written as \( x = 12k + 8 \) for some integer \( k \). - Now, check if \( x \) is divisible by 4: \( x = 12k + 8 = 4(3k + 2) \). - Since \( x \) can be expressed as \( 4 \) times an integer, \( x \in T \). 4. **Proving \( T \not\subseteq S \)**: - Not every element of \( T \), which is divisible by 4, leaves a remainder of 8 when divided by 12. - For example, \( 4 \in T \) but \( 4 \mod 12 = 4 \neq 8 \), so \( 4 \notin S \). #### Problem 6 Let \( S = \{ x \in \mathbb{Z} \mid \exists r \in \mathbb{Z}, x = 12r + 7 \} \) and \( T = \{ x \in \mathbb{Z} \mid \exists s \in \mathbb{Z}, x = 3s - 2 \} \). **Task**: Prove that \( S \subseteq T \), but \( T \not\subseteq S \). #### Solution Steps: 1. **Understanding \( S \)**: - The set \( S \)
Transcribed Image Text:### Problems involving Subsets and Modular Arithmetic #### Problem 5 Let \( S = \{ x \in \mathbb{Z} \mid x \mod 12 = 8 \} \) and \( T = \{ x \in \mathbb{Z} \mid 4 \mid x \} \). **Task**: Prove that \( S \subseteq T \), but \( T \not\subseteq S \). #### Solution Steps: 1. **Understanding \( S \)**: - The set \( S \) consists of all integers \( x \) that, when divided by 12, leave a remainder of 8. 2. **Understanding \( T \)**: - The set \( T \) consists of all integers \( x \) that are divisible by 4. 3. **Proving \( S \subseteq T \)**: - Every element \( x \in S \) can be written as \( x = 12k + 8 \) for some integer \( k \). - Now, check if \( x \) is divisible by 4: \( x = 12k + 8 = 4(3k + 2) \). - Since \( x \) can be expressed as \( 4 \) times an integer, \( x \in T \). 4. **Proving \( T \not\subseteq S \)**: - Not every element of \( T \), which is divisible by 4, leaves a remainder of 8 when divided by 12. - For example, \( 4 \in T \) but \( 4 \mod 12 = 4 \neq 8 \), so \( 4 \notin S \). #### Problem 6 Let \( S = \{ x \in \mathbb{Z} \mid \exists r \in \mathbb{Z}, x = 12r + 7 \} \) and \( T = \{ x \in \mathbb{Z} \mid \exists s \in \mathbb{Z}, x = 3s - 2 \} \). **Task**: Prove that \( S \subseteq T \), but \( T \not\subseteq S \). #### Solution Steps: 1. **Understanding \( S \)**: - The set \( S \)
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