Show that if A and B, (n = 1,2,3, . -) are events, then An (U Bn) = U (AN Bm)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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![### Problem 15
**Problem Statement:**
Show that if \( A \) and \( B_n \) (\( n = 1, 2, 3, \ldots \)) are events, then
\[ A \cap \left( \bigcup_{n=1}^{\infty} B_n \right) = \bigcup_{n=1}^{\infty} (A \cap B_n) \]
**Explanation:**
- **Events \( A \) and \( B_n \):** These are sets representing particular outcomes in a probability space.
- **Union of Sets:** \( \bigcup_{n=1}^{\infty} B_n \) represents the union of all events \( B_n \) from \( n=1 \) to infinity. It includes all outcomes that are in at least one of the \( B_n \) events.
- **Intersection of Sets:** \( A \cap \left( \bigcup_{n=1}^{\infty} B_n \right) \) represents the set of outcomes that are both in event \( A \) and in any of the \( B_n \) events.
- **Right-hand Side:** \( \bigcup_{n=1}^{\infty} (A \cap B_n) \) represents the union of intersections, meaning it includes outcomes that are in both \( A \) and a specific \( B_n \) for some \( n \).
**Conceptual Understanding:**
The equation asserts the distributive property of intersections over infinite unions. It states that intersecting a set \( A \) with a union of other sets \( B_n \) is equivalent to taking the union of intersections of \( A \) with each \( B_n \). This is a fundamental property in set theory used extensively in probability and analysis.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe6156c38-2816-48c7-aed8-10dafecc3b80%2F4de619f9-d324-4065-8b60-797464df3e95%2F5w1ztk9_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem 15
**Problem Statement:**
Show that if \( A \) and \( B_n \) (\( n = 1, 2, 3, \ldots \)) are events, then
\[ A \cap \left( \bigcup_{n=1}^{\infty} B_n \right) = \bigcup_{n=1}^{\infty} (A \cap B_n) \]
**Explanation:**
- **Events \( A \) and \( B_n \):** These are sets representing particular outcomes in a probability space.
- **Union of Sets:** \( \bigcup_{n=1}^{\infty} B_n \) represents the union of all events \( B_n \) from \( n=1 \) to infinity. It includes all outcomes that are in at least one of the \( B_n \) events.
- **Intersection of Sets:** \( A \cap \left( \bigcup_{n=1}^{\infty} B_n \right) \) represents the set of outcomes that are both in event \( A \) and in any of the \( B_n \) events.
- **Right-hand Side:** \( \bigcup_{n=1}^{\infty} (A \cap B_n) \) represents the union of intersections, meaning it includes outcomes that are in both \( A \) and a specific \( B_n \) for some \( n \).
**Conceptual Understanding:**
The equation asserts the distributive property of intersections over infinite unions. It states that intersecting a set \( A \) with a union of other sets \( B_n \) is equivalent to taking the union of intersections of \( A \) with each \( B_n \). This is a fundamental property in set theory used extensively in probability and analysis.
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