Question

Does {1,2,3} ,{3,4,5}, {1,4}, {1,5}, {2,4}, {2,5} form an incidence geometry? If so do any of the parallel postulates hold (Elliptic, Euclidean, Hyperbolic parallel postulates)?

Answer 1

Yes,{1,2,3} ,{3,4,5}, {1,4}, {1,5}, {2,4}, {2,5} form an incidence geometry.

think of a ractangle with its two diagonal then put 3 at point of intersection of the diagonal and other 1,2,4,5 at vetices(with 1,2 diagonally opposite and 4,5 diagonally opposite).This is compatible with the given set of lines and points.

Incidence Geometry Incidence Axiom

1 For every pair of points P and Q there exists a unique line l incident with P and Q.

2 . For every line l there exist at least two distinct points incident with l.

3 . There exist three distinct points A, B, C not simultaneously incident with a common line l.

these all are satisfied clearly.

Euclidean Parallel Postulate- Through a point not on a line there is exactly one line parallel to the given line.

Hyperbolic Parallel Postulate-Through a point not on a line there is more than one line parallel to the given line.

Elliptic Parallel Postulate-Any two lines intersect in at least one point.

It is not following any one of Elliptic Parallel Postulate , Hyperbolic Parallel Postulate and Euclidean Parallel Postulate.