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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

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  • $\begingroup$ What did you try? $\endgroup$ – Patrick Da Silva Jan 25 '14 at 9:55
  • $\begingroup$ I have three points and three lines pass through the three points so far. $\endgroup$ – bsdshell Jan 25 '14 at 9:59
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    $\begingroup$ 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? $\endgroup$ – Patrick Da Silva Jan 25 '14 at 10:22
  • $\begingroup$ yep, four points and six lines, thx $\endgroup$ – bsdshell Jan 25 '14 at 20:13
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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.

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  • $\begingroup$ "If K is any field, the vector space V=K2 has always the structure of affine plane" any idea how to prove that? $\endgroup$ – bsdshell Jan 25 '14 at 20:14
  • $\begingroup$ well, the smallest field is $\mathbb{Z}_{2}$ $\endgroup$ – bsdshell Jan 25 '14 at 22:00
  • $\begingroup$ @bsdshell: Define the lines to be the translates of the $1$-dimensional subspaces. Check the axioms! $\endgroup$ – Andrea Mori Jan 25 '14 at 22:42

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