**Question:**

In a flat geometry, a set of points E is called convex, if for each pair of points $ M, N \in E $, the segment MN is contained in Ea) Prove that the intersection of the two convex sets is a convex set. b) Is the union of two convex sets a convex set?

**Answer**

a) Assume F, G are convex sets and $ F \cap G \neq \varnothing $

Consider the points $ M, N (M \neq N) $ in $ F \cap G $ we have

$ M \in F \cap G \Rightarrow M \in F $ because F is a convex set so $ MN \subset F (1) $

$ N \in F \cap G \Rightarrow N \in F $

Similar to $ MN \subset G (2) $

From (1) and (2) deduce $ MN \subset F \cap G: F \cap G $ is a convex set

b) Cannot be confirmed

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