Solutions to Problem Set 4
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The distance between two vertices in a graph is the length of the shortest … The diameter of the graph above is 5. The mostdistant vertices are A and G, …
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Massachusetts Institute of Technology 6.042J/18.062J, Fall ‘05: Mathematics for Computer Science October 12 Prof. Albert R. Meyer and Prof. Ronitt Rubinfeld revised October 10, 2005, 616 minutes
Solutions to Problem Set 4
Problem 1. For functions f?:?AB?and g?:?BC, the composition of g?and f, written
??
g?. f, is the function h?:?AC?where
?
h(a)::=?g(f(a)).
?
(a)
Prove that if f?and g?are bijections, then so is g?. f.
(b)
Prove that if f?:?AB?is a bijection, then there is a bijection, e?:?BA?such that
?
e?. f?=?IA, where IA?:?A?? A?and IA(a)::=?a?for all a?? A. ?
(c) Prove that graph isomorphism is an equivalence.
Problem 2. The proof of the Handshake Theorem in Week 5 Notes is a little more informal
than is desirable in the beginning of 6.042. Rewrite the proof more carefully as an induction on the number of edges in a graph.
Problem 3. The distance between two vertices in a graph is the length of the shortest path between them. For example, the distance between two vertices in a graph of airline connections is the minimum number of flights required…
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