MIT6_042JF10_rec08

MIT6_042JF10_rec08 - 6.042/18.062J Mathematics for Computer...

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6.042/18.062J Mathematics for Computer Science October 6, 2010 Tom Leighton and Marten van Dijk Problems for Recitation 8 1 Build-up error Recall a graph is connected iff there is a path between every pair of its vertices. False Claim. If every vertex in a graph has positive degree, then the graph is connected. 1. Prove that this Claim is indeed false by providing a counterexample. 2. Since the Claim is false, there must be a logical mistake in the following bogus proof. Pinpoint the Frst logical mistake (unjustiFed step) in the proof. Proof. We prove the Claim above by induction. Let P ( n ) be the proposition that if every vertex in an n -vertex graph has positive degree, then the graph is connected. Base cases : ( n 2). In a graph with 1 vertex, that vertex cannot have positive degree, so P (1) holds vacuously. P (2) holds because there is only one graph with two vertices of positive degree, namely, the graph with an edge between the vertices, and this graph is connected. Inductive
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MIT6_042JF10_rec08 - 6.042/18.062J Mathematics for Computer...

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