Lecture Notes 1: Affine Planes

Defn: An affine plane is an ordered pair (, ) where is a nonempty set of elements called points and a nonempty collection of subsets of called lines which have the following properties:
A1: If P and Q are distinct points, there is a unique line l such that P l and Q l. [This line is denoted l(P,Q).]
A2: If P is a point not contained in the line l, there is a unique line m such that P m and m l = . (When l m = , l is said to be parallel to m, written l || m.)
A3: There are at least two points on each line; there are at least two lines.
  (This non-degeneracy condition can be replaced with : There exist three points not on the same line.)
An axiom system is consistent if the axioms are not self-contradictory, that is, the assumption of the truth of the axioms will not lead to a contradiction. If an axiom system has a model (i.e., an example) in which all the axioms hold, then it is consistent. The following examples each show that the axioms for an affine plane are consistent.

Examples:

The Real Coordinate Plane
= {(x,y): x,y }
and
l in iff l = {(x,y): ax + by = c}, where a,b,c , a2 + b2 0. (This last condition is just a fancy way to say that both of a and b can not be 0 simultaneously)
The Rational Affine Plane
= {(x,y): x,y }
and
l iff l = {(x,y): ax + by = c}, where a,b,c , a2 + b2 0.
The Smallest Affine Plane
= {A, B, C, D }
and
= {{AB}, {AC}, {AD}, {BC}, {BD}, {CD}}.
One can also view this example as a coordinate plane over the finite field Z2 (= GF(2)) by setting A = (0,0), B = (1,0), C = (0,1) and D = (1,1).
To see that these examples really do give affine planes we prove some theorems concerning coordinate planes defined over fields. In the coordinate planes, lines are defined as the sets of points whose coordinates (x,y) satisfy a linear equation, i.e., ax + by = c with both a and b not simultaneously zero. We note that if (x,y) satisfies such an equation then it also satisfies max + mby = mc for any m in the field. Therefore, we see that the "equation" which defines a line is not unique, i.e., the coefficients may be different and yet the equations define the same line. However, equations which define the same line always have coefficients that are scalar multiples of each other. That is to say, ax + by = c and a'x + b'y = c' define the same line if and only if there is a non-zero m in the field so that a' = ma, b' = mb and c' = mc. We will use this information in the next theorem.

Theorem 1: In any field coordinate plane, any two distinct points are on a unique common line.

Pf: Let P = (x1,y1) and Q = (x2,y2) be distinct points.
We will first show that these two points are on a common line.
Case I: Suppose that x1 = x2.
Then the coordinates of P and Q satisfy the equation x = x1. That is, the equation ax + by = c with a = 1, b = 0 and c = x1.
Case II: Suppose that x1x2.
Let a = y2- y1, b = x1- x2 and c = (y2-y1)x1+ (x1-x2)y1, then we claim that P and Q both satisfy the equation

ax + by = c.
That P satisfies the equation is obvious, so consider
(y2-y1)x2 + (x1-x2)y2 = y2x2 - y1x2 + x1y2 - x2y2
= (y1x1 - x1y1) - y1x2 + x1y2
= y1(x1 - x2) + x1(y2 - y1)
= (y2 - y1)x1 + (x1 - x2)y1.

We now wish to show that the common line is unique.

In either of the two cases above, we have shown that P and Q satisfy an equation of the form ax + by = c. Thus we have the system,

ax1 + by1 = c
ax2 + by2 = c.
We can subtract these equations to obtain a(x1-x2) + b(y1-y2) = 0.
Now, if x1 = x2, we have y1y2 (since P and Q are distinct points) and so, b = 0. The equation satisfied by P and Q thus looks like ax = c with a0, and we have ax1 = c. So, ax = ax1, or x = x1, and this line is uniquely determined.

If x1x2, we can solve our equation for a to get, a = b(y2-y1)(x1-x2)-1 = bm. We now have, ax1+ by1 = bmx1+ by1 = b(mx1+ y1) = c. Therefore, any line determined by P and Q has an equation of the form bmx + by = b(mx1+ y1). So, if these points were also on the line a'x + b'y = c', then we would have b'mx + b'y = b'(mx1+ y1) and we would have a' = a(b'/b), b' = b(b'/b) and c' = c(b'/b) and the lines are the same.

Proposition 2: Two distinct lines in a field coordinate plane are parallel if and only if they have equations of the form ax + by = c and ax + by = d, where cd.

Pf: If cd then there is no point (x1,y1) so that c = ax1 + by1 = d, so there is no point in common on the two lines and they are parallel.
On the other hand, the simultaneous equations

ax + by = c,
a'x + b'y = c',
will have a common solution unless (the determinant) ab' - ba' = 0. There are two cases to consider. If a = 0, then since b0, we must have a' = 0. The equations are thus by = c and by = c'(b/b') = d (which we get by multiplying both sides of the second equation by b/b'). Since the lines are distinct cd. Otherwise, a0, and b' = ba'/a. The equations now become ax + by = c and a'x + ba'/a y = c'. Multiplying the second by a/a' gives ax + by = c'a/a' = d, and this is distinct from the first line if and only if cd.

Theorem 3: In a field coordinate plane, given a point P not on a line l, there is a unique line containing P and parallel to l.

Pf: Suppose that point P = (p1,p2) is not on the line l with equation ax + by = c. Then, cap1 + bp2. Any line parallel to l has an equation of the form ax + by = d with cd. If such a line is to contain P then d = ap1 + bp2. So, the only line containing P and parallel to l will have the equation ax + by = ap1 + bp2.

Theorems 1 and 3 show that axioms A1 and A2 are satisfied by any field coordinate plane and all of our examples are field coordinate planes. It is straightforward to check that A3 is also satisfied in all of our examples.

An axiom system is independent if no axiom can be proved from the remaining axioms of the system. To show that a set of axioms is independent, we find models in which all the axioms except one are valid, for each axiom in the system. Independence is a desirable mathematical property for an axiom system, but it may not be a pedagogically useful property. Often one uses non-independent axioms systems to simplify the presentation of material or the level of difficulty of the subject matter. The following three examples, taken together, provide a proof of the independence of the the affine plane axiom system.

Independence Examples:

Independence of A1
= {A, B, C, D }
and
= {{AB}, {CD}}.
There is no line joining B and C, so axiom A1 is not satisfied. On the other hand, A2 and A3 are satisfied as can easily be checked.
Independence of A2 (the Fano Plane)
The seven points and seven lines of the Fano plane satisfy Axioms A1 and A3 but not A2 (point 1 is not on line {345}, but there is no line through 1 which is parallel to {345}.)

A simpler example with the same properties is given by:
= {A, B, C }
and
= {{AB}, {BC}, {AC}}.

Independence of A3
= {A, B }
and
= {{AB}}.
Axiom A1 is clearly satisfied. Axiom A2 is satisfied vacuously, that is, there is no instance of the axiom being false since the hypotheses of the axiom are never satisfied. Since there is only one line, axiom A3 is not satisfied.