soc 375 - exams - 2005

soc 375 - exams - 2005 - Sociology 375 Exam 1 Spring 2005...

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Sociology 375 Exam 1 Spring 2005 Prof Montgomery Answer all questions. 200 points possible. 1) [40 points] Consider a relation R on a set S. Slightly modifying the definition given in lecture, suppose that R is complete iff ¬ (xRy yRx) (x=y) for all (x,y) S × S . a) For each of the following conditions, state whether the condition is equivalent to the definition of completeness given above, and then explain why (or why not) using truth tables. [HINT: To save time, you might construct one large truth table to answer all 3 parts simultaneously.] i) xRy yRx (x=y) for all (x,y) S × S ii) (xRy yRx) ¬ (x=y) for all (x,y) S × S iii) xRy yRx for all distinct (x,y) S × S b) Consider the relation R = {(1,1), (1,2), (2,2), (2,3), (3,2)} on S = {1, 2, 3}. Using the definition of completeness given above, determine whether R is complete by testing all ordered pairs. [NOTE: To receive full credit, you must demonstrate that you have tested all ordered pairs. You can show this by constructing a table in which the rows correspond to ordered pairs, the columns correspond to propositions in the definition of completeness, and the entries in the table are truth values.] 2) [60 points] Consider relation R = {(1,4), (2,1), (2,3), (2,4), (4,3)} on S = {1, 2, 3, 4}. a) Show how the relation R can be represented as (i) an adjacency matrix A (ii) a (directed) graph b) In general, given a relation represented as directed graph, how can you tell (using the graph) whether the relation is asymmetric? In the present example, is the relation R asymmetric? If not, list any violations of the asymmetry condition. c) State the matrix test for transitivity, and then show whether A passes this test. If not, list any violations of the transitivity condition. d) Is R a strict partial order? If it is, then draw the Hasse diagram. If not, find the transitive closure of R, and then draw the Hasse diagram of the transitive closure. e) Using the adjacency matrix A, compute the Bonacich centrality vector given α = 1 and arbitrary β > 0. [HINT: There are no 4-paths in A.]
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3) [25 points] Consider an arbitrary adjacency matrix A. a) State the formula for the reachability matrix (given adjacency matrix A). b) For each of the properties below, indicate whether the reachability relation will always , sometimes , or never have this property. If you answer sometimes , then explain what property of the A matrix would determine the outcome. i) reflexive ii) symmetric iii) transitive c) Given the answer to part (b), what condition(s) on the A matrix determine whether reachability is an equivalence relation? What do social network analysts call the equivalence classes of the reachability relation? 4) [15 points]
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soc 375 - exams - 2005 - Sociology 375 Exam 1 Spring 2005...

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