What is the definition of convex as you are applying it?
Prove: Every ray is convex.
Here is what I think:
Let's say we have a union of a triangle and the set of all points that lie inside that triangle called A. If we place the points P,Q in there, then A becomes convex since the entire segment PQ lies in A. And since it is known that eveery segemnt PQ is a convex set, and that a set with only one point is convex, then a ray, say ray AB, ic convex also because it contains two points so it has the property of having two points.
Is this right? If it isn't, can someone please help me with it? I really do not understand how else to prove it.
Thank you for your time and effort! I really appreciate it!
A set A is called convex if for every two points P, Q of A, the entire segment PQ lies in A.
For example, every segment PQ is a convex set. In fact, a set with only one point is convex. ( Since such a set does not contain any two points, it follows that every two points of it have any property we feel like mentioning)
This all comes straight from out of my text book.
Thanks for the help!
Let the point of emanation of the ray be point O. Suppose for a point C in between A and B on the line segment AB that the ray does not pass through C. OA and OB are the same ray, but OC cannot be the same ray as OA, because then the ray would pass through C. This would put OC at a positive angle away from OA, and OCB and ACB would be triangles. But ACB cannot be a triangle because C is on AB. Hence OC is on OA, and for all points C between A and B, OC is the same ray as OA and C is on that ray. So the definition is satisfied.