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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
- Computer Science