soc 375 - exams - spring 2007

soc 375 - exams - spring 2007 - Sociology 375 Exam 1 Spring...

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Sociology 375 Exam 1 Spring 2007 Prof Montgomery Answer all questions. 250 points possible. Explanations can be brief. 1) [40 points] Given a binary vector v (with elements equal to 0 or 1, which may be interpreted as truth values), we could use matlab to compute the truth values of the following eight expressions: all(v), all(~v), ~all(v), ~all(~v), any(v), any(~v), ~any(v), ~any(~v). [HINT: Recall that ~ is the logical “not” operator in matlab. Thus, the truth values in the vector v are reversed in the vector ~v, and the truth value of all(v) is reversed by ~all(v).] Assuming that the binary vector v has only 2 elements (each of which could be either “true” or “false”), use a truth table to determine which of the preceding eight expressions are logically equivalent (producing the same answer regardless of the truth values in v). [HINT: Because v has only two elements, the vector v could be written as v = [P Q] where P and Q are truth values, the all command becomes the logical “and” operator, and the any command becomes the logical “or” operator.] To receive full credit, you must show the truth table and partition the eight expressions into equivalence classes. 2) [90 points] Consider the relation A (‘sends information to’) on a set of four actors S = {1,2,3,4}. In particular, suppose that 1 sends information to 2, 2 sends information to 1, 2 sends information to 3, 3 sends information to 4, and 4 sends information to 3. a) Show how the relation A could be represented as i) a set ii) a directed graph iii) an adjacency matrix b) Compute i) the number of 3-paths between each ordered pair of actors ii) the reachability matrix iii) the distance matrix c) In this example, is the reachability relation an equivalence relation? If so, find the equivalence classes generated by this relation. If not, explain why. d) Give the adjacency matrix for the relation ‘can be reached by.’ Forming the intersection of the relations ‘can reach’ and ‘can be reached by’, we obtain the relation ‘can reach and be reached by’. Is this relation an equivalence relation? If so, find the equivalence classes generated by this relation. If not, explain why . e) Suppose that we partition the set of actors into those actors who can be reached by all others and those actors who cannot. Which actors are in the first subset? Which are in the second subset? Is this partition a regular equivalence? If not, briefly explain why.
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3) [50 points] Consider an Olympic tournament with teams from America (A), Britain (B), Canada (C) and Denmark (D). In the first round, A beat B, and C beat D. There was then a “gold medal” game in which A beat C, and a “bronze medal” game in which B beat D. a) Give the adjacency matrix for the ‘beat’ relation.
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soc 375 - exams - spring 2007 - Sociology 375 Exam 1 Spring...

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