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3. AnBCC. Prove the following statement: For all sets A, B, and C, if BCC, then

3. AnBCC. Prove the following statement: For all sets A, B, and C, if BCC, then

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:** 3. Prove the following statement: For all sets \( A, B, \) and \( C \), if \( B \subseteq C \), then \( A \cap B \subseteq A \cap C \). --- **Explanation:** The problem involves a proof in set theory. It states a relationship involving three sets \( A, B, \) and \( C \), and their intersections. The goal is to show that if set \( B \) is a subset of set \( C \), then the intersection of \( A \) and \( B \) is a subset of the intersection of \( A \) and \( C \). The statement leverages a fundamental property of sets: the intersection of two sets. **Definitions:** - **Subset \( B \subseteq C \):** This means every element of set \( B \) is also an element of set \( C \). - **Intersection \( A \cap B \):** This is a set containing all elements that are both in set \( A \) and set \( B \). **Proof Outline:** 1. **Assume:** Let \( x \) be an element of \( A \cap B \). By definition of intersection, \( x \) must be in both \( A \) and \( B \). 2. **Subset Condition:** Since \( B \subseteq C \), the element \( x \), which is in \( B \), must also be in \( C \). 3. **Intersection Conclusion:** Given \( x \) is in both \( A \) (from step 1) and \( C \) (from step 2), \( x \) is in \( A \cap C \). 4. **Subsetting Result:** Since any element \( x \) in \( A \cap B \) can also be found in \( A \cap C \), it follows that \( A \cap B \subseteq A \cap C \). By following these steps, one can logically demonstrate the proof required in the problem statement.
Transcribed Image Text:**Problem Statement:** 3. Prove the following statement: For all sets \( A, B, \) and \( C \), if \( B \subseteq C \), then \( A \cap B \subseteq A \cap C \). --- **Explanation:** The problem involves a proof in set theory. It states a relationship involving three sets \( A, B, \) and \( C \), and their intersections. The goal is to show that if set \( B \) is a subset of set \( C \), then the intersection of \( A \) and \( B \) is a subset of the intersection of \( A \) and \( C \). The statement leverages a fundamental property of sets: the intersection of two sets. **Definitions:** - **Subset \( B \subseteq C \):** This means every element of set \( B \) is also an element of set \( C \). - **Intersection \( A \cap B \):** This is a set containing all elements that are both in set \( A \) and set \( B \). **Proof Outline:** 1. **Assume:** Let \( x \) be an element of \( A \cap B \). By definition of intersection, \( x \) must be in both \( A \) and \( B \). 2. **Subset Condition:** Since \( B \subseteq C \), the element \( x \), which is in \( B \), must also be in \( C \). 3. **Intersection Conclusion:** Given \( x \) is in both \( A \) (from step 1) and \( C \) (from step 2), \( x \) is in \( A \cap C \). 4. **Subsetting Result:** Since any element \( x \) in \( A \cap B \) can also be found in \( A \cap C \), it follows that \( A \cap B \subseteq A \cap C \). By following these steps, one can logically demonstrate the proof required in the problem statement.
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