Question

A projective plane is a plane (S,L ) satisfying the following four axioms. P1. For any two distinct points P and Q there is one and only one line containing P and Q. P2. For any two distinct lines l and m there exists one and only one point P belonging to l∩m. P3. There exist three noncollinear points. P4. Every line contains at least three points.

a)Give an example of a plane (S,L) that satisfies axioms P1, P2 and P3 but not P4.

b) Give an example of a plane (S,L) that satisfies axioms P1, P2 and P4 but not P3.

c) Give an example of a plane (S,L) that satisfies axioms P1, P3 and P4 but not P2.

d) Give an example of a plane (S,L) that satisfies axioms P2, P3 and P4 but not P1.

Answer 1