MIT6_042JS10_lec10_sol

MIT6_042JS10_lec10_sol - Massachusetts Institute of...

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Massachusetts Institute of Technology 6.042J/18.062J, Spring ’10 : Mathematics for Computer Science February 24 Prof. Albert R. Meyer revised February 24, 2010, 756 minutes Solutions to In-Class Problems Week 4, Wed. Problem 1. Direct Prerequisites Subject 18.01 6.042 18.01 18.02 18.01 18.03 8.01 8.02 8.01 6.01 6.042 6.046 18.02, 18.03, 8.02, 6.01 6.02 6.01, 6.042 6.006 6.01 6.034 6.02 6.004 (a) For the above table of MIT subject prerequisites, draw a diagram showing the subject num- bers with a line going down to every subject from each of its (direct) prerequisites. Solution. Creative Commons 2010, Prof. Albert R. Meyer .
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2 Solutions to In-Class Problems Week 4, Wed. (b) Give an example of a collection of sets partially ordered by the proper subset relation, , that is isomorphic to (“same shape as”) the prerequisite relation among MIT subjects from part ( a ). Solution. For each subject, S , let preset ( S ) ::= S | S is an indirect prerequisite of S OR S = S . For example, subject preset 18 . 02 { 18 . 01 , 18 . 02 } 18 . 03 { 18 . 01 , 18 . 03 } 6 . 006 { 6 . 042 , 18 . 01 , 6 . 01 , 8 . 01 , 6 . 006 } Note that the “ OR S = S ” clause is necessary: if we let the set representing subject S just be the indirect prerequisites of S , then 18.02 and 18.03, for example, would be represented by the same set, { 18 . 01 } . Then the correspondence between subjects and sets would no longer be a bijection,
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