sets-final-takehome - Here are two connected graphs The...

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NAME: Takehome problem Sets and Logic Final Exam Answer the problem stated below. Staple any extra sheets to the back of this one, and bring them to the exam on Monday April 26 at 3:00 pm to be handed in with the inclass exam. Sign the following statement: On my honor as a University of Florida student, I have not received help from or otherwised discussed this problem with anyone except my instructor, nor have I consulted books other than the textbook for help with this problem. Signed: Problem Use induction to prove the following theorem. For the purposes of this question, a graph is a finite set of points in the plane, some of which are connected by straight line segments, none of which intersect. A graph is connected if you can get from any point of the graph to any other by going on the line segments. Theorem. Suppose G is a connected graph. Then v G + r G = e G + 2 (1) where v G is the number of points. r G is the number of regions (counting the infinite outside area as one region). e G is the number of line segments. Example.
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Unformatted text preview: Here are two connected graphs. The single dot has one point, one region, and no lines, so since 1 + 1 = 0 + 2 it satisfies equation (1). The other has 4 points, 2 regions and 4 lines, so equation (1) correctly asserts 4 + 2 = 4 + 2. Your proof may use the following fact: Fact. If G is any connected graph, then at least one of the following is true: 1. G has only one point (and so has no lines and one region). 2. G has at least one edge with one free end, so that removing that edge and the free point will still leave a connected graph (with the same regions as before). 3. G has an edge which can be deleted, without removing any points, leaving a connected graph and merging two regions of the original graph into one region. In the example with 4 points, the upper line is an example of type 2. Any of the other three lines are examples of type 3: removing one of them will merge the region inside the triangle with the region outside, leaving only one region....
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