Question

Part II. (4 pts) Given the axiom set for the Incidence Geometry as below: Undefined terms: point, line, on Definitions: 1. Two lines are intersecting if there is a point on both. 2. Two lines are parallel if they have no point in common. Axioms: I. Given any two distinct points, there is a unique line on both. II. Each line has at least two distinct points on it. III. There exist at least three points. IV. Not all points lie on the same line. Consider the following axiom set for a geometry (let's refer this geometry as Geometry A): Undefined terms: point, line, incidence Axioms I. Given any two distinct points, there is a unique line incident with both. Il. Given any two distinct lines, there is a unique point incident with both. III. There exists in this geometry a set of four points, no three of which are on the same line. (Warning, this doesn't imply that there are exactly four points in this geometry.) IV. Each line contains exactly n points. a. (1.5 pts) Does this axiom set establish an incidence geometry? Please justify for each axiom. b ( 2.5 pts) Determine whether each of the following statements is TRUE (T) or FALSE (F) in Geometry A. No justification necessary. Please circle T for True and F for False. T F (1) in Geometry A, every pair of distinct lines has at least one point in common. T F (2) in Geometry A, each point has at least three lines on it. T F (3) Geometry A has the Euclidean parallel property T F (4) Geometry A has three lines that don't share a common point. T F (5) In Geometry A, n23. 2

Answer 1