Hey guys, was hoping for some input on my proof:
Q- Prove that in an affine plane of order n each line contains exactly n points.
Axioms:
A1: At least 4 distinct points, no three are collinear
A2: At least one line with exactly n points on it
A3: Given 2 distinct points, there is exactly one line incident with both of them
A4: Given a point A and a line r, not through A, there is at most one line through A which does not meet r.
Pf: We prove that in an affine plane of order n each line contains exactly n points. Given an affine plane of order 2 there will be 4 distinct points which form a square. By axiom 2 we see that there exists at least one line with exactly 2 points on it. But axiom 3 states that each pair of distinct points must have a line incident with each of them. Therefore, each pair of points will have a line between them which has exactly two points on it. Axiom 4 is satisfied as our model is a square and will have two sets of parallel lines QED.
Am I approaching these types of geometric proofs correctly?