Question

1) Assume that (P. L. d) satisfies postulates 1-6 of neutral geometry. Let C P be a ne. We denote by H and H the two sides of (see postulate 6) (a) Show that both sides H+ and H are non-empty. (20 points) (b) Show that both sides H+ and H are infinite sets. (10 points)

Answer 1

Let $l \subset \mathbb{P}$ be a line in the plane given. Now are two sides of the line.

a) if any of the side is empty then $l \subset \mathbb{P}$ will act as a boundary for $\mathbb{P}$. But as per definition of a plane, planes have no boundaries, it extends infinitely to all sides. Thus either of cannot be empty.

b) suppose one side of line $l \subset \mathbb{P}$ have only finite number of points. Consider the point in that side which is the farthest point from the line $l \subset \mathbb{P}$ . Now consider the line passing through parallel to line . Then there exist no point of $\mathbb{P}$ in one side of the line , hence will be a bound for $\mathbb{P}$, which is a contradiction. Hence both $H_+,H_-$ contains infinite number of points.