Two models M and M' of Incidence Geometry (i.e., of the three incidence axioms) are said to be isomorphic if there is a one-to-one correspondence between their points and between their lines such that a point P in M lies on a line l in M iff the corresponding point P' in M' lies on the corresponding line l' in M'.

1. Let M be the following interpretation: points: A, B, C, D; lines: {A, B}, {A, C}, {A, D}, {B, C, D}. Is M a model of Incidence Geometry? Prove your answer.
2. Let M be the following interpretation: points: A, B, C; lines: {A, B}, {A, C}, {A, B, C}. Is M a model of Incidence Geometry? Prove your answer.
3. Give a model of Incidence Geometry with four points and six lines. Does your model have three collinear points? Do you think every model of Incidence Geometry  that has four points and six lines is isomorphic to your model? Explain all your answers.
4. Let M and M' be models of Incidence Geometry.  Let A, B, C be points in M, and A', B', C' their corresponding points in M'. Prove that A, B, C are collinear iff A', B', and C' are collinear.
5. Let M and M' be models of Incidence Geometry. Let l and m be lines in M, and let l' and m' be their corresponding lines in M'. Prove or disprove: l || m iff l' || m'.
6. Prove that the Euclidean plane and the Poincare' Disk Model (PDM) are not isomorphic models of Incidence Geometry. You may assume that the Euclidean plane satisfies EPP, and the PDM satisfies HPP. Hint: Use one or more of the above problems.
7.  Let S be the set of all incidence and betweenness axioms (seven axioms altogether). Then, two models M and M' of S are said to be isomorphic if there is a one-to-one correspondence between their points and between their lines such that "the correspondence preserves the relations of incidence and betweeness," i.e.:

   1.  a point P in M lies on a line l in M iff the corresponding point P' in M' lies on the corresponding line l' in M', and
   2. if A, B, C are in M and A', B', C' are their corresponding points in M' respectively, then A*B*C iff A'*B'*C'.

   Let A, B be points and l a line in M, with A', B', l' their corresponding points and lines in M'. Prove or disprove: A and B are on the same side of l iff A' and B' are on the same side of l'.