4. Prove that if A and B are convex sets, then AN B is also convex.

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Author:Erwin Kreyszig
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**Problem 4: Convex Sets Intersection**

**Statement:**

Prove that if \( A \) and \( B \) are convex sets, then \( A \cap B \) is also convex.

**Explanation:**

A set is considered convex if, for any two points within the set, the line segment connecting these two points also lies entirely within the set. To prove that the intersection of two convex sets \( A \) and \( B \) is convex, consider any two points \( x \) and \( y \) in the intersection \( A \cap B \). Since \( x \) and \( y \) are in both \( A \) and \( B \), and both \( A \) and \( B \) are convex by assumption, the line segment connecting \( x \) and \( y \) will be entirely contained within both \( A \) and \( B \). Consequently, it is also contained within \( A \cap B \), proving that the intersection is convex.
Transcribed Image Text:**Problem 4: Convex Sets Intersection** **Statement:** Prove that if \( A \) and \( B \) are convex sets, then \( A \cap B \) is also convex. **Explanation:** A set is considered convex if, for any two points within the set, the line segment connecting these two points also lies entirely within the set. To prove that the intersection of two convex sets \( A \) and \( B \) is convex, consider any two points \( x \) and \( y \) in the intersection \( A \cap B \). Since \( x \) and \( y \) are in both \( A \) and \( B \), and both \( A \) and \( B \) are convex by assumption, the line segment connecting \( x \) and \( y \) will be entirely contained within both \( A \) and \( B \). Consequently, it is also contained within \( A \cap B \), proving that the intersection is convex.
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