
Re: A question about straight lines
Posted:
Jan 28, 2014 2:11 PM


Neighbor says: >About the "open ray": how can a finite straightline have no boundary???? Being made of points, there must be two boundaries in both directions, which we call endpoints, for if there is no boundary this is not a finite straightline but an infinite one.
You are correct that the "boundary" is the same point for both AG and GB, but that boundary point "G" can be considered either included (the "closed" situation) or excluded ("open") in either AG or GB. Being a boundary and belonging to the set to which it is a boundary are separate notions, and one does not follow logically from the other.
If you want the two segments to overlap at G, then you set theoretically consider it "included" in both AG & GB. (There is no logical conflict in including the number "2" in two distinct sets, say { 1, 2 } and { 1, 2, 3 }  same here.) If you want the two segments to be disjoint, then one or the other or both need to be open. If you want to recover the segment AB by "unioning" the two fragments set theoretically and also keep AG & GB disjoint, then one has to be open and the other closed. This is all elementary set theory applied to line segments considered as sets of point.
Cheers, Joe N

