**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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Answered: 4. Prove that if A and B are convex…

Math

Advanced Math

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

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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Concept explainers

Set Theory

Every field in the present world involves a set. The theory of sets was developed by German mathematician Georg Cantor while working on the “problems of trigonometric series”.

Question

**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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