Advance Geometry

Math 356
1. Consider the following axioms for Four-Line Geometry:
A1. There exist exactly four lines.
A2. Any two distinct lines are on exactly one point.
A3. Every point is on exactly two lines.
(a) Prove that this system is independent.
(b) Define two points to be parallel if they are not on the same line. Prove the following theorem in
this system: For each point P, there is exactly one point parallel to P.
Note: Do not assume properties of this geometry based solely on its model.
2. For a fixed integer n > 1, here are the axioms for Finite Affine Plane of order n:
A1. There exist at least four distinct points, no three of which are collinear.
A2. There exists at least one line with exactly n points on it.
A3. Given two distinct points, there is exactly one line containing both of them.
A4. Given a line ` and a point P not on `, there is exactly one line through P that is parallel to `.
Prove that in an affine plane of order n, each line ` has exactly n