soc 375 - exams - 2006

soc 375 - exams - 2006 - Sociology 375 Exam 1 Spring 2006...

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Sociology 375 Exam 1 Spring 2006 Prof Montgomery Answer all questions. 200 points possible. Explanations can be brief. 1) [20 points] Some textbooks specify the condition for asymmetry as xRy ¬ yRx while other textbooks specify the condition as ¬ (xRy yRx) Use a truth table to determine whether these conditions are equivalent. 2a) [5 points] What is the mathematical definition of a relation? b) [5 points] Suppose you were told to form the union of two partial orders. Does this instruction make sense? Briefly explain why or why not. c) [5 points] Suppose you were told to find the equivalence classes generated by a partial order. Does this instruction make sense? Briefly explain why or why not. 3) Consider a set of students S = {Al, Beth, Carl, David, Ellen}. You learn that Al likes Beth, Al likes Carl, Beth likes Carl, Carl likes Beth, David likes Beth, Ellen likes Al, and Ellen likes Carl. a) [20 points] Show how the relation R (‘likes’) on the set S could be represented (i) as a set (ii) using infix notation (iii) as a (directed) graph (iv) as an adjacency matrix b) [25 points] Does the relation R on S satisfy each of the following conditions? If not, you should identify one violation. [HINT: You need to report only one violation.] (i) reflexivity (ii) antireflexivity (iii) symmetry (iv) antisymmetry (v) transitivity
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4) Consider a relation R on a set of four individuals S = {1, 2, 3, 4}. This relation is characterized by the following adjacency matrix: 0 1 0 0 A = 1 0 1 1 0 0 0 1 0 0 1 0 a) [40 points] Given the adjacency matrix A, compute the following (in matrix form): (i) number of 2-paths (ii) number of 3-paths (iii) reachability (iv) distance b) [15 points] Is the reachability relation (from question 4.a.iii) of the following type? Briefly note why or why not. (i) equivalence relation (ii) quasiorder (iii) partial order c) [10 points] State the condition for structural equivalence. Given the relation above (characterized by the adjacency matrix A), are there any individuals who are structurally equivalent to each other? If so, identify those individuals. If not, briefly explain why. d) [20 points] State the condition for regular equivalence. Given the relation above (characterized by the adjacency matrix A), can you partition S into equivalence classes (characterized by an adjacency matrix E) such that individuals 3 and 4 are regularly equivalent to each other? If so, verify that E satisfies the regular equivalence condition. Otherwise, briefly explain why individuals 3 and 4 cannot be regularly equivalent. 5) Consider a softball league with 5 teams. Over the course of the season, each team played each other team 3 times. Last season’s outcomes are given by the “beats” matrix below where B(i,j) is the number of times team i beat team j. 0
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soc 375 - exams - 2006 - Sociology 375 Exam 1 Spring 2006...

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