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Unformatted text preview: Massachusetts Institute of Technology 6.042J/18.062J, Spring ’11 : Mathematics for Computer Science March 7 Prof. Albert R Meyer revised Monday 7 th March, 2011, 05:20 Problem Set 5 Due : March 14 Reading: Chapter 9 – 9.10.1 , Parallel Task Scheduling. Skip Chapter 9.11 , Equivalence Relations and Chapter 10 , Communication Nets, which will not be covered this term. Problem 1. How many binary relations are there on the set f 0;1 g ? How many are there that are transitive?, ...asymmetric?, ...reflexive?, ...irreflexive?, ...strict partial orders?, ...weak partial orders? Hint: There are easier ways to find these numbers than listing all the relations and checking which prop erties each one has. Problem 2. The following procedure can be applied to any digraph, G : 1. Delete an edge that is in a cycle. 2. Delete edge h u ! v i if there is a path from vertex u to vertex v that does not include h u ! v i . 3. Add edge h u ! v i if there is no path in either direction between vertex u and vertex v . Repeat these operations until none of them are applicable. This procedure can be modeled as a state machine. The start state is G , and the states are all possible digraphs with the same vertices as G . (a) Let G be the graph with vertices f 1;2;3;4 g and edges fh 1 ! 2 i ; h 2 ! 3 i ; h 3 ! 4 i ; h 3 ! 2 i ; h 1 ! 4 ig What are the possible final states reachable from G ?...
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This note was uploaded on 10/04/2011 for the course CSCI 101 taught by Professor Leighton during the Spring '11 term at MIT.
 Spring '11
 Leighton
 Computer Science

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