MATH 520 Axiomatic Systems and Finite Geometry Proofs
Name ________________________
This assignment is due 1 week from today on Wednesday, September 5 at the beginning of class in hard copy.
Typed submissions (double spaced, Times New Roman 12) will receive a 5 point bonus. Handwritten
submissions (double spaced) must be neat and easily readable. I have posted this assignment in Word format so
that you can save it and then simply type.
Undefined Terms and Axioms for Fano’s Geometry:
Undefined Terms: point, line, on
Axiom 1. There exists at least one line.
Axiom 2. There are exactly three points on every line.
Axiom 3. Not all points are on the same line.
Axiom 4. There is exactly one line on any two distinct points.
Axiom 5. There is at least one point on any two distinct lines.
Fano’s Theorem 1: In Fano’s geometry, two distinct lines have exactly one point in common.
Proof:
Fano’s Theorem 2a: Fano’s geometry contains exactly seven points.
Proof:
Fano’s Theorem 2b: Fano’s geometry contains exactly seven lines.
Proof:
Fano’s Theorem 3: Each point is on exactly three lines.
Proof:
Fano’s Theorem 4: The set of lines on any point contains all the points of the geometry.
Proof: