I What are the properties of a plane cut into regions by a set of lines?

[Mr Davis 97]

Problem: Let $L$ be a set of $n$ lines in the plane in general position, that is, no three of them containing the same point. The lines of $L$ cut the plane into $k$ regions. Prove by induction on $n$ that this subdivision of the plane has $\binom{n}{2}$ vertices, $n^2$ edges, and $\binom{n}{2} + n + 1$ cells.

I don't need help solving this problem, I just need help interpreting it. What does it mean that the plane is cut into $k$ regions? I thought that the number of regions was determined by $n$. Also, what's the point of the $k$ if we're not proving anything about it?

Finally, what is meant by cells? Also, are edges the finite segments between intersections?

 

[BvU]

What does it mean that the plane is cut into $k$ regions

Draw two lines: you get 4 regions, 1 vertex
Draw three lines: you get 7 regions, 3 vertices
Draw another line: you get 11 regions, 6 vertices
You see $k$ back in the number of 'cells'

edges the finite segments between intersections

some of them are finite, some infinite

My main tip: make a few sketches -- the question becomes clear and the answer becomes clear as well.