HW3CS40

# HW3CS40 - not list all the characteristics that you can...

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Homework “Structures”, CS40, Summer 2008 15 points total due Wednesday, September 3, 11:00 am Question 1 (Relations on pairs of integers, 3 points) . Consider the set S = Z × Z and deﬁne the relation R S × S by R = { (( a , x ) , ( b , y )) : a b and x y } . Prove that R is a partial ordering, but not a total ordering. Question 2 (Functions, 3 points) . Is it possible for a function f : S S to be surjective but not bijective? Prove your answer. Question 3 (Drawing Graphs, 3 points) . Let V be the set of binary strings of length 3 , that is: V = { 000 , 001 , ... , 111 } . Deﬁne the undirected graph G = ( V , E ) with the edges E = { ( x , y ) : x and y differ by exactly one bit } . Draw G in a way that reveals the structure of G as clearly as possible. Question 4 (Equivalences vs. Graphs, 3 points) . Let E be an equivalence relation on the ﬁnite set V . What can you say about the corresponding graph G = ( V , E ) ? Note: do
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Unformatted text preview: not list all the characteristics that you can think of, but rather describe in a few sentences the necessary and sufﬁcient properties of such graphs. Question 5 (Spanning Graphs, 3 points) . Let G = ( V , E ) be a connected, undirected graph. We say that H is a “spanning graph” of G if and only if H = ( V , F ) is also a connected, undirected graph, with F ⊆ E . Prove that the smallest possible spanning graph, i.e. the one with the smallest possible set of edges F , will always be a tree. Note: this question tries to test your skills in interpreting by yourself new deﬁnitions (like the one here for spanning graphs). As a result, the TA (PK) and I (WvD) will not help you with this speciﬁc exercise....
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