Math 367 Homework Assignment 1
due Thursday, January 28
For the exercises below, refer to the following incidence axioms:
(I1) For every pair of distinct points P and Q there exists exactly one line ` such that both
P and Q lie on `.
(I2) For every line ` there exist at least two distinct points P and Q such that both P and
Q lie on `.
(I3) There exist three points that do not all lie on any one line.
1. (The real line.) Interpret point to mean any point on the number line, and interpret line
to mean the set of all points. Which of the above incidence axioms is satisfied?
2. (Closed intervals.) Interpret point to mean any point on the number line, and interpret
line to mean any closed interval [a, b] where a and b are real numbers with a < b. (Notation:
the interval [a, b] is defined to be the set of all real numbers x for which a ≤ x ≤ b.) Which of
the above incidence axioms is satisfied?
3. (Four line geometry.) There are 6 points, labeled A, B, C, D, E, F . There are 4 lines, as
indicated in the picture.
•F
• E
D •
•
•
•
A
B
C
(a) Which of the above incidence axioms is satisfied? (b) Can this model be modified by
including additional lines to ensure that all axioms are satisfied? If so, list the additional
lines. If not, explain why not. (Remember that not all lines need contain the same number of
points.)
4. (Etch-a-sketch geometry.) A point is any ordered pair (x, y) of real numbers (i.e. a
point in the Cartesian plane). A line is any continuous path in the Cartesian plane consisting
entirely of horizontal and vertical line segments. For example, the picture below is of such a
line.
Which of the above incidence axioms is satisfied?
5. (Finite affine plane of order 3.) Look up (e.g. on the web) the finite affine plane of
order 3. There should be 9 points and 12 lines. (a) Sketch a picture indicating all 9 points
and 12 lines. (b) Label the 9 points as A, B, C, D, E, F, G, H, I and list, in set notation, the
12 lines. (For example, {A, B, C} denotes the line containing exactly the points A, B, and C.)
(c) Are the three incidence axioms above all satisfied?