Euclidean and non-Euclidean Geometry – Fall 2008
Dr. Hamblin
Section 2.3 Worksheet: Checking Axioms
Recall the five incidence axioms:
I-1.
Each two distinct points determine a line.
I-2.
Three noncollinear points determine a plane.
I-3.
If two points lie in a plane, then any line containing those two points lies in that plane.
I-4.
If two distinct planes meet, their intersection is a line.
I-5.
Space consists of at least four noncoplanar points, and contains three noncollinear points.
Each plane is a set of points of which at least three are noncollinear, and each line is a set of
at least two distinct points.
Exercises
1. Determine which of the five incidence axioms are satisfied in this model.
Points: {1, 2, 3, 4}
Lines: {1, 2, 3}, {1, 4}, {2, 3}, {1, 3, 4}
Planes: {1, 2, 3}, {2, 3, 4}
2. Construct a model that shows that axiom I-3 is independent from the other axioms.