1. **Problem Statement:** We are given an affine plane with axioms I1 to I4 and need to prove four properties about parallel lines, points on lines, total points, and provide an example. 2. **(i) Transitivity of Parallelism:** Given lines $\ell_1 \parallel \ell_2$ and $\ell_2 \parallel \ell_3$, show $\ell_1 \parallel \ell_3$. - By axiom I4, for any line and a point not on it, there is a unique parallel line through that point. - Since $\ell_1 \parallel \ell_2$, and $\ell_2 \parallel \ell_3$, if $\ell_1$ met $\ell_3$, then $\ell_2$ would intersect $\ell_3$ contradicting $\ell_2 \parallel \ell_3$. - Hence, $\ell_1$ does not meet $\ell_3$, so $\ell_1 \parallel \ell_3$. 3. **(ii) Equal Number of Points on Any Two Lines:** - Let $\ell$ and $m$ be any two lines. - By axiom I3, there exist points not all on the same line, so lines have at least two points (I2). - For each point on $\ell$, axiom I4 guarantees a unique parallel line through that point parallel to $m$. - This correspondence between points on $\ell$ and points on $m$ is one-to-one. - Therefore, $|\ell| = |m|$. 4. **(iii) Total Number of Points if a Line has $n$ Points:** - Suppose a line $\ell$ has exactly $n$ points. - By axiom I4, through each point not on $\ell$, there is a unique parallel line to $\ell$. - There are $n$ such parallel lines, each with $n$ points (from (ii)). - These lines partition the plane into $n$ lines each with $n$ points. - Total points $= n \times n = n^2$. 5. **(iv) Example of an Affine Plane with 4 Points:** - Consider the affine plane over the field $\mathbb{F}_2$ (field with 2 elements). - Points: $\{(0,0), (0,1), (1,0), (1,1)\}$. - Lines: sets of points with equations $x=0$, $x=1$, $y=0$, $y=1$, $x+y=0$, $x+y=1$. - Each line has exactly 2 points, total points are $2^2=4$. **Final answers:** (i) $\ell_1 \parallel \ell_3$ (ii) Any two lines have the same number of points. (iii) Total points in the plane $= n^2$ if a line has $n$ points. (iv) Example: affine plane over $\mathbb{F}_2$ with 4 points.