For Aand B subsets of formal A x CBU C) is a se t x iprovide subset
Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
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![**Title: Set Theory: Subsets and Proofs**
**Introduction**
In set theory, understanding how subsets interact with union and intersection operations is a fundamental concept. Here, we explore a specific subset relationship and provide a formal proof.
**Problem Statement**
For \( A \) and \( B \), subsets of the universal set \( X \), provide a formal proof for:
\[ A \times (B \cup C) \text{ is a subset of } (A \times B) \cup (A \times C) \]
**Proof Outline**
1. **Definitions and Notation:**
- \( A \times (B \cup C) \): Cartesian product of \( A \) with the union of \( B \) and \( C \).
- \( (A \times B) \cup (A \times C) \): Union of the Cartesian products of \( A \) with \( B \) and \( A \) with \( C \).
2. **Approach:**
- To prove that for every element \((a, b) \in A \times (B \cup C)\), it must also be in \((A \times B) \cup (A \times C)\).
- Consider an arbitrary element \((a, y) \in A \times (B \cup C)\).
3. **Verification:**
- If \( y \in B \), then \((a, y) \in A \times B\).
- If \( y \in C \), then \((a, y) \in A \times C\).
- Therefore, \((a, y)\) is in \((A \times B) \cup (A \times C)\).
**Conclusion**
This proof demonstrates the subset relationship by showing that an arbitrary element of the set on the left-hand side must exist in the set on the right-hand side. This is consistent with set theory principles concerning unions and Cartesian products.
**Reflection:**
Understanding these relationships aids in comprehending more complex concepts in set theory and provides a base for further exploration in mathematics.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7dbb4ae4-0d65-4baa-9481-63f79be91eca%2F30c775af-c04b-4363-ae11-899c62df7e54%2Fqqcm2w6.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Set Theory: Subsets and Proofs**
**Introduction**
In set theory, understanding how subsets interact with union and intersection operations is a fundamental concept. Here, we explore a specific subset relationship and provide a formal proof.
**Problem Statement**
For \( A \) and \( B \), subsets of the universal set \( X \), provide a formal proof for:
\[ A \times (B \cup C) \text{ is a subset of } (A \times B) \cup (A \times C) \]
**Proof Outline**
1. **Definitions and Notation:**
- \( A \times (B \cup C) \): Cartesian product of \( A \) with the union of \( B \) and \( C \).
- \( (A \times B) \cup (A \times C) \): Union of the Cartesian products of \( A \) with \( B \) and \( A \) with \( C \).
2. **Approach:**
- To prove that for every element \((a, b) \in A \times (B \cup C)\), it must also be in \((A \times B) \cup (A \times C)\).
- Consider an arbitrary element \((a, y) \in A \times (B \cup C)\).
3. **Verification:**
- If \( y \in B \), then \((a, y) \in A \times B\).
- If \( y \in C \), then \((a, y) \in A \times C\).
- Therefore, \((a, y)\) is in \((A \times B) \cup (A \times C)\).
**Conclusion**
This proof demonstrates the subset relationship by showing that an arbitrary element of the set on the left-hand side must exist in the set on the right-hand side. This is consistent with set theory principles concerning unions and Cartesian products.
**Reflection:**
Understanding these relationships aids in comprehending more complex concepts in set theory and provides a base for further exploration in mathematics.
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