Axiomatic Geometry - Mathematics 360 - Spring 2009

For Homework 21

1. Use the axioms below to prove for every point there is a line not incident with it.
   Undefined terms: point, line, incident with.
   Axiom 1: Every two points are incident with exactly one line.
   Axiom 2: Every line is incident with at least two points.
   Axiom 3: There are at least three points that are not all incident with the same line.
2. Use the Scorpling Flugs axioms to prove if A scorples B, then B does not scorple A.
3. Let A, B, C be three distinct flugs. Prove that it is possible to rename A, B, C in such a way that A scorples B and B scorples C.
4. Suppose we take "flug" to mean "set" and take "scorple" to mean "is a subset of." Let A, B, C, D be the following flugs (sets):
   1. A = {1, 2}; B = {1, 3}, C = {1, 2, 3},  D = {1, 2, 3, 4}. Is this "interpretation" a model for the Scorpling Flugs axiom system, i.e., does this interpretation satisfy all four axioms?
   2. A = {1}; B = {1, 2}, C = {1, 2, 3},  D = {1, 2, 3, 4}. Is this interpretation a model for the Scorpling Flugs axiom system?
   3. Can you think of a different model (not using sets and subsets) for the Scorpling Flugs axiom system?