MIT6_042JS10_lec03_sol - Massachusetts Institute of...

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Massachusetts Institute of Technology 6.042J/18.062J, Spring ’10 : Mathematics for Computer Science February 8 Prof. Albert R. Meyer revised January 26, 2010, 73 minutes Solutions to In-Class Problems Week 2, Mon. Problem 1. The proof below uses the Well Ordering Principle to prove that every amount of postage that can be paid exactly using only 6 cent and 15 cent stamps, is divisible by 3. Let the notation “ j | k indicate that integer j is a divisor of integer k , and let S ( n ) mean that exactly n cents postage can be paid using only 6 and 15 cent stamps. Then the proof shows that S ( n ) IMPLIES 3 | n, for all nonnegative integers n. (*) Fill in the missing portions (indicated by “. . . ”) of the following proof of (*). Let C be the set of counterexamples to (*), namely 1 C ::= { n | ... } Solution. n is a counterexample to (*) if n cents postage can be made and n is not divisible by 3, so the predicate S ( n ) and NOT (3 | n ) defines the set, C , of counterexamples. Assume for the purpose of obtaining a contradiction that C is nonempty. Then by the WOP, there is a smallest number, m C . This m must be positive because. . . . Solution. . . . 3 | 0 , so 0 is not a counterexample. But if S ( m ) holds and m is positive, then S ( m 6) or S ( m 15) must hold, because. . . . Solution.
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This note was uploaded on 05/27/2011 for the course CS 6.042J taught by Professor Prof.albertr.meyer during the Spring '11 term at MIT.

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MIT6_042JS10_lec03_sol - Massachusetts Institute of...

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