O Show that if Si and S2 are convex sets in R™X", then so is t mxn S = {(x, y1 + y2) | x € R", y1, y2 € R", (x, y1) e
O Show that if Si and S2 are convex sets in R™X", then so is t mxn S = {(x, y1 + y2) | x € R", y1, y2 € R", (x, y1) e
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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It should be Rm+n instead of Rmxn
![## Convex Sets and Their Partial Sums
### Problem 2.16
**Objective:**
Show that if \( S_1 \) and \( S_2 \) are convex sets in \( \mathbb{R}^{m \times n} \), then their partial sum is also convex.
**Partial Sum Defined as:**
\[ S = \{ (x, y_1 + y_2) \mid x \in \mathbb{R}^m, y_1, y_2 \in \mathbb{R}^n, (x, y_1) \in S_1, (x, y_2) \in S_2 \} \]
### Explanation
1. **Understanding Convex Sets:**
- A set \( C \) in a real vector space is convex if, for any two points \( a \) and \( b \) in \( C \), the line segment joining \( a \) and \( b \) is also in \( C \).
- Mathematically expressed, if \( \lambda \in [0, 1] \), then \( \lambda a + (1 - \lambda) b \in C \).
2. **Given Conditions:**
- \( S_1 \) and \( S_2 \) are convex sets in \( \mathbb{R}^{m \times n} \).
3. **Challenge:**
- Prove that the set \( S \) formed by the described operation is also convex.
4. **Approach to Solution:**
- Verify if any linear combination of elements from \( S \) remains in \( S \).
- Consider arbitrary elements \( a = (x, y_1 + y_2) \) and \( b = (x', y_1' + y_2') \) in \( S \).
- Show that for any \( \lambda \in [0, 1] \), the combination \( \lambda a + (1 - \lambda) b \) is also in \( S \).
### Conclusion
By following these steps and verifying the properties, the convexity of the set \( S \) as defined can be confirmed. This demonstrates how the operation on two convex sets can lead to another convex set, maintaining the property of convexity.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F718b1378-40e4-4c32-83bc-211fc46d7de4%2F2eee8f22-724e-44fc-a5e6-a90866561a82%2Fc6vca3_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Convex Sets and Their Partial Sums
### Problem 2.16
**Objective:**
Show that if \( S_1 \) and \( S_2 \) are convex sets in \( \mathbb{R}^{m \times n} \), then their partial sum is also convex.
**Partial Sum Defined as:**
\[ S = \{ (x, y_1 + y_2) \mid x \in \mathbb{R}^m, y_1, y_2 \in \mathbb{R}^n, (x, y_1) \in S_1, (x, y_2) \in S_2 \} \]
### Explanation
1. **Understanding Convex Sets:**
- A set \( C \) in a real vector space is convex if, for any two points \( a \) and \( b \) in \( C \), the line segment joining \( a \) and \( b \) is also in \( C \).
- Mathematically expressed, if \( \lambda \in [0, 1] \), then \( \lambda a + (1 - \lambda) b \in C \).
2. **Given Conditions:**
- \( S_1 \) and \( S_2 \) are convex sets in \( \mathbb{R}^{m \times n} \).
3. **Challenge:**
- Prove that the set \( S \) formed by the described operation is also convex.
4. **Approach to Solution:**
- Verify if any linear combination of elements from \( S \) remains in \( S \).
- Consider arbitrary elements \( a = (x, y_1 + y_2) \) and \( b = (x', y_1' + y_2') \) in \( S \).
- Show that for any \( \lambda \in [0, 1] \), the combination \( \lambda a + (1 - \lambda) b \) is also in \( S \).
### Conclusion
By following these steps and verifying the properties, the convexity of the set \( S \) as defined can be confirmed. This demonstrates how the operation on two convex sets can lead to another convex set, maintaining the property of convexity.
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