Day01 - CS 314 Day 1 Item#1 Proofs formal and informal...

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Page 1 of 3 CS 314 Day 1: January 22, 2008 Item #1. Proofs formal and informal, Mathematics. Theorem: A set of n elements has 2 n subsets. Proof (Exercise 1, p. 13): First, an informal proof. Let each subset be represented by a binary number with n digits. For example, for the set {A, B, C}, the number 101 would represent the subset {A, C}, where 1 means A is in, 0 means B is out, and 1 means C is in. For a set with n elements, all possible binary numbers of length n represent all possible subsets. They range from this to this hence there this many subsets: 3 2 1 n 0 ... 000 3 2 1 n 1 ... 111 , but that’s 2 n in binary. 3 2 1 n 0 ... 000 1 Why is this merely an informal proof? Convert it into a formal proof. Item #2. Proofs formal and informal, Computer Science. Theorem: Two computers are not any more powerful than one computer in terms of functions that can be computed. Proof: Create a computer C that works as follows. Let it run 1 minute duplicating as much computation as computer A can do in 1 min.; then let it run 1 minute duplicating as much computation as computer B can do in 1 min. In general, in minutes 2n and 2n+1, C is doing the work of A and B in minute n. Thus as time goes on, everything that A and B compute is also computed by C. Why is this merely an informal proof? What are some problems that need to be addressed to convert it to a more formal proof? Item #3. Induction can be tricky. “Describe the fallacy in the following ‘proof’ by induction:
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This note was uploaded on 04/19/2008 for the course COMP 314 taught by Professor Chase during the Spring '08 term at Dickinson.

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Day01 - CS 314 Day 1 Item#1 Proofs formal and informal...

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