How to interpret a line equation in 4-point geometry (affine plane of order 2). I am currently reading "Basic Notions of Algebra" by Igor Shafarevich. In the first chapter example of a coordinatization of 4-point geometry is given.
Set of axioms:


*

*Through any two distinct points there is one and only one line.

*Given any line and a point not on it, there exists one and only one other line through the point and not intersecting the line (that is, parallel to it).

*There exist three points not on any line.



In this geometry we have 4 points A, B, C, D and 6 lines AB, CD; AD, BC; AC, BD. The families of parallel lines are separated by semicolons.
Let $\Bbb{0,1}$ be symbols with operations $+$ and $\times$ such that
$$
\begin{array}{cc}
\begin{array}{c|cc}
\text{+} & 0 & 1\\
\hline
0 & 0 & 1\\
1 & 1 & 0\\
\end{array}
&
\begin{array}{c|cc}
\times & 0 & 1\\
\hline
0 & 0 & 0\\
1 & 0 & 1\\
\end{array}
\end{array}
$$
The pair of quantities 0 and 1 with operations defined on them as above serve us in coordinatising the "geometry". For this, we give points coordinates (X, Y) as follows: A = (0, 0), B = (0, 1), C = (1, 0), D = (1, 1).
It is easy to check that the lines of the geometry are then defined by the linear equations:
$$
\begin{array}{ccc}
& AB: 1X = 0;      & CD: 1X = 1; & AD: 1X + 1Y = 0;\\
& BC: 1X + 1Y = 1; & AC: 1Y = 0; & BD: 1Y = 1;\\
\end{array}
$$
The question is: how does one should interpret this equations?
Any suggestions will be appreciated.
 A: So, now the base field, i.e. the number line instead of all the real numbers, consists only of $0$ and $1$, with $1+1=0$.
It is stated that the plane over this $2$ element field has six lines. Later those equations are just the equations of these 6 lines.
For instance, we have $A=(0,0)$ and $B=(0,1)$, and the line $AB$ is the one with equation
$$X=0$$
and indeed, these two points satisfy this equation. 
Or, take $AD$ with $D=(1,1)$, both points satisfy $X+Y=0$. 
And so on.
A: The equations describe the lines, that is, if a point satisfies the equation then it is on the line.  For example the line $X = 0$ is satisfied by all points $(X,Y)$ with $X = 0$ (with the coordinates $X$ and $Y$ coming from $\mathbb{F}_{2}$, the finite field of order 2).  This gives an algebraic way to describe the lines.
You can think of the lines as having a slope of either $0$, $1$, or $\infty$ ($Y$ coefficient divided by $X$ coefficient), these slopes determine your parallel classes. This should let you go from a pair of points to the equation by determining a slope and then using a point to get the remaining value to describe the line as $AX+BY=C$.
