I Interpretation of a problem

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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

Science Advisor
Homework Helper
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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.
 
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