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lecture8 - Lectures 8 and 9 Proofs (cont) Announcements...

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Lectures 8 and 9 Proofs (cont)

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Announcements This week Hw2 due on Friday Hw3 out on Friday We will finish Chapter 1 and start Chapter 2
Recap Three fundamental proof techniques Direct proof Proof by construction Proof by contraposition Proof by contradiction

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Last lecture: Theorem If P is a u-v shortest path on a graph, P does not contain any cycles
Last lecture: Theorem If n > 2 is even, there exists a graph with n vertices where all the vertices have degree 3

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Proof by construction We prove the theorem by showing that such a graph exists for n=4,6,8, … For each n = 2k, k>2 we can construct the following graph: 1 2 3 k k+1 k+2 2k In this graph, the vertices are ordered cyclically on a circle. Each vertex is connected to the vertex right before, right after and right across. That is: For i=1, …, 2k we add the edges (i, (i+1)) In addition, for i=1, …, k, we add the edge (i, i+k). This completes the construction. Note that we use the cyclic order (i.e. 2k+1 = 1) Clearly, for i=1, …, k, vertex i has exactly three neighbors: i-1, i+1 and i+k For i=k+1, … , 2k, vertex i will also have three neighbors: i-1, i+1 and i-k
Proof by construction Note that we needed to construct an example for each n=4,6,8, … Therefore, we came up with a systematic way of constructing such graphs, and presented the details of the construction

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Prove or Disprove Every positive integer is the sum of the squares of two integers After trying for a while… Hmm, maybe the theorem is not true Let’s try to find a counter example
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This note was uploaded on 10/21/2011 for the course CSCI 2011 taught by Professor Staff during the Spring '08 term at Minnesota.

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lecture8 - Lectures 8 and 9 Proofs (cont) Announcements...

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