PROVE There exist exactly n2 n lines The following are the

PROVE: There exist exactly n² + n lines.

The following are the axioms for a finite affine plane of order n > 1: Axiom 1 There exist at least four distinct points no three of which axe collinear. Axiom 2 There exists at least one line incident with exactly n points. Axiom 3 Each two points are incident with exactly one line. Axiom 4 Given a point P not on a line l, there exists exactly one line incident with P parallel to l.

Solution

Let l be a line of n² matrix . Then any point l has n + 1 lines through it . If we throw out l itself, this gives that through each point of there are n lines that are not parallel to l . This shows there are exactly n² lines of n² matrix that are not parallel to l . This leaves the lines that are parallel to l and by property that each line has exactly n lines parallel to it , therefore there are n such lines . Hence , total number of lines are n² + n .