In each of the following interpretations of the undefined terms, which of the ax

Question

In each of the following interpretations of the undefined terms, which of the axioms of incidence geometry are satisfied and which are not? Tell whether each interpretation has the elliptic, Euclidean, or hyperbolic parallel property. "Points" are lines in Euclidean three-dimensional space, "lines" are planes in Euclidean three-space, "incidence" is the usual relation of a line lying in a plane. Same as in part (a), except that we restrict ourselves to lines and planes that pass through a fixed point O.

Explanation / Answer

(a) It is Euclidean Property because, it is given that  a "line" (plane in 3D) and a "point" (line) not on it, there is only one "line" (plane) that does not intersect the given one - which means the plane that is parallel in 3D. so the correct choice would be Euclidean.

(b) It is Elliptic Property because, if each "line" (plane) goes through O, then each "line" (plane) has an intersection with every other "line" (plane), as they must have O in common. So either the twowill be or they will intersect properly.

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