# what is the minimum number of points in affine plane.

what is minimum number of points in affine plane,

By the way: Here are the $\textbf{Three Axioms}$ for affine plane.

1. Given two distinct points $\textbf{P}$ and $\textbf{Q}$, there is only one line passing through them

2. Given a point $\textbf{P}$ and a line $\textit{l}$, if $\textbf{P}\not\in\textit{l}$, there is only one line passing through point $\textbf{P}$ and parallel to line $\textit{l}$

3. There exist three points $\textbf{P}$, $\textbf{Q}$, $\textbf{R}$ non-collinear

• What did you try? – Patrick Da Silva Jan 25 '14 at 9:55
• I have three points and three lines pass through the three points so far. – bsdshell Jan 25 '14 at 9:59
• But three points do not form an affine plane because 1 tells you it has to be a triangle, but 2 tells you the triangle doesn't work. What about four points? – Patrick Da Silva Jan 25 '14 at 10:22
• yep, four points and six lines, thx – bsdshell Jan 25 '14 at 20:13

HINT: If $K$ is any field, the vector space $V=K^2$ has always the structure of affine plane.
Now take for $K$ the smallest field you know.
• well, the smallest field is $\mathbb{Z}_{2}$ – bsdshell Jan 25 '14 at 22:00
• @bsdshell: Define the lines to be the translates of the $1$-dimensional subspaces. Check the axioms! – Andrea Mori Jan 25 '14 at 22:42