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MIT6_042JS10_lec01_sol

# MIT6_042JS10_lec01_sol - Massachusetts Institute of...

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Unformatted text preview: 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 Solutions to 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. Solution. log x < , for < x < 1 , so since both sides of the inequality “ 3 > 2 ” are being multiplied by the negative quantity log 10 (1 / 2) , the “ > ” in the second line should have been “ < .” (b) Bogus proof : 1 ¢ = \$0 . 01 = (\$0 . 1) 2 = (10 ¢ ) 2 = 100 ¢ = \$1 . Solution. \$0 . 01 = \$(0 . 1) 2 = (\$0 . 1) 2 because the units \$ 2 and \$ don’t match (just as in physics the difference between sec 2 and sec indicates the difference between acceleration and velocity). Similarly, (10 ¢ ) 2 = 100 ¢. (c) Bogus Claim : If a and b are two equal real numbers, then a = 0 . Bogus proof. a = b a 2 = ab a 2 − b 2 = ab − b 2 ( a − b )( a + b ) = ( a − b ) b a + b = b a = . 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. 2 Solutions to In-Class Problems Week 1, Wed....
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MIT6_042JS10_lec01_sol - Massachusetts Institute of...

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