Interpretation # 9

Points:	A, B, C, D, and E.
Lines:	{A, B}, {A, C}, {A, D}, {A, E}, {B,C}, {B, D} {B, E},{C, D}, {C, E}, and {D, E}.
Incidence:	PIl if P is an element of l.

QUESTIONS:

Does this interpretation satisfy the incidence axiom I-1?

Does this interpretation satisfy the incidence axiom I-2?

Does this interpretation satisfy the incidence axiom I-3?

Is this interpretation a model?

Determine whether this interpretation has the elliptic, Euclidean, of hyperbolic parallel property.

ANSWERS:

Does this interpretation satisfy the incidence axiom I-1?

Yes, if P and Q are any of the letters A, B, C, D and E, {P, Q} is the unique line in which they both lie.

Does this interpretation satisfy the incidence axiom I-2?

Yes, every line in this interpretation has exactly two points, therefore at least two.

Does this interpretation satisfy the incidence axiom I-3?

Yes, points A, B, and C are distinct and non-collinear.

Is this interpretation a model?

Yes, all three incidence axioms I-1, I-2, and I-3 are satisfied.

Determine whether this interpretation has the elliptic, Euclidean, of hyperbolic parallel property.

This interpretation has the hyperbolic parallel property, because for any line {P, Q}, and any point R that is not on {P, Q} (therefore R is different from P and Q), there exist two lines {R, S}, and {R, T} through R and parallel to {P, Q}. ( S , and T are points that the sets {A, B, C, D, E} and  {P, Q, R, S, T} are the same). For example there exist two lines that go through A and are parallel to the line {B, D} (lines {A, C}, and {A, E})