To continue, we need tools to examine the projective plane we work with, like circles in ancient Greek geometry. We introduce the most fundamental of those tools here, but before we define it, we need a few definitions.
The set of all points incident to a given line is the [math]\textbf{range}[/math] of that line.
The set of all lines incident to a given point is the [math]\textbf{pencil}[/math] of that line.
In a projective plane, let ranges and pencils be [math]\textbf{incidence sets}[/math].
The incidence sets of the point p and line L can be written as In(,L) and In(p,) respectively.
Before we explicitly define this tool, we demonstrate it with the proof of the following theorem:
Theorem 2.1: Any two distinct ranges from the same plane have the same cardinality.
Proof: We observe that every line has a unique range and vice versa, thus the two resultant lines, which we call A and B, are also distinct. By Axiom 5*, we have a point o not incident to either A or B. Now, consider the following function.
[eqn]\begin{align*}
Pe : In(,\textup{A}) &\to In(,\textup{B})\\
\textup{a} &\mapsto \textup{B[oa]}.
\end{align*}[/eqn]
The construction of the points B[o?] for points on A is shown in the pic.
If d and e are any distinct points on A, then the lines od and oe are distinct lines which meet at o. Since o does not lie on B, then by Axiom 1, the points B[od] and B[oe] are distinct. Since d and e were arbitrary, then Pe is injective.
Now take an arbitrary point x on B. Let z = A[ox]. The point z exists and lies on A, so Pe(z) exists. {z, o, x} is collinear and x lies on B, so Pe(z) = B[oz] = x, thus Pe is surjective.
Since Pe is surjective and injective, it is a bijection between In(,A) and In(,B), thus they have the same cardinality. #
The function used in the proof above is the tool we seek, which we will formally define in the next large post.
