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MIT6_042JS10_lec01_prob

MIT6_042JS10_lec01_prob - Massachusetts Institute of...

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Massachusetts Institute of Technology 6.042J/18.062J, Fall ’09 : Mathematics for Computer Science February 3 Prof. Albert R. Meyer revised January 25, 2010, 1076 minutes In-Class Problems Week 1, Wed. Problem 1. Identify exactly where the bugs are in each of the following bogus proofs. 1 (a) Bogus Claim : 1 / 8 > 1 / 4 . Bogus proof. 3 > 2 3 log 10 (1 / 2) > 2 log 10 (1 / 2) log 10 (1 / 2) 3 > log 10 (1 / 2) 2 (1 / 2) 3 > (1 / 2) 2 , and the claim now follows by the rules for multiplying fractions. (b) Bogus proof : 1 ¢ = $0 . 01 = ($0 . 1) 2 = (10 ¢ ) 2 = 100 ¢ = $1 . (c) Bogus Claim : If a and b are two equal real numbers, then a = 0 . Bogus proof. a a 2 a 2 b 2 ( a b )( a + b ) a + b a = = = = = = b ab ab b 2 ( a b ) b b 0 . Creative Commons 2010, Prof. Albert R. Meyer . 1 From Stueben, Michael and Diane Sandford. Twenty Years Before the Blackboard , Mathematical Association of Amer- ica, ©1998.
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2 In-Class Problems Week 1, Wed. Problem 2. It’s a fact that the Arithmetic Mean is at least as large the Geometric Mean, namely, a + b ab 2 for all nonnegative real numbers a and b . But there’s something objectionable about the following
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