hw3sol - CS123 Spring 2010 Solution to Homework 3 Problem 1...

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CS123 Spring 2010 Solution to Homework 3 Problem 1 (20 points) For each of the following 4 relations on the set L of all living people on January 1, 2010, state (no proof necessary) if the relation is reflexive, symmetric, antisymmetric, transitive, equivalence relation, and/or partial order. In the case of equivalence relations, explain what every equivalence class signifies. In the case of partial orders, indicate if the set L has a minimum, a maximum, minimals, and/or maximals, and explain what minimum, maximum, minimal and maximal mean in that context. a) x R y if x is taller than y or x is the same person as y. Solution: This relation is reflexive, not symmetric, antisymmetric, transitive and is a partial order. In the case that we have only one tallest (shortest) person in the given set L, this person is the minimum (maximum). The tallest person is also the minimal and the shortest person is also the maximal. If there are more than one person who tie for the tallest (shortest), then the set L does not have a minimum (maximum), but does have minimals (maximals), which are these sets of people. b) x R y if x and y have the same first name. Solution: This relation is reflexive, symmetric, not antisymmetric (practically speaking), transi- tive, and is an equivalent relation. The equivalence classes are the sets of people having the same first name. c) x R y if x graduated from the same university as y. Solution: This relation is not reflexive, symmetric, not antisymmetric (practically speaking), transitive, and is neither an equivalence relation nor a partial order. d) x R y if the annual income of x is less than that of y or x is the same person as y. Solution: This relation is reflexive, not symmetric, antisymmetric, transitive and is a partial order. In the case that we have only one person who has the most (least) annual income in the given set L, this person is the maximum (minimum). The person with the most annual income is also the maximal; The person with the least annual income is also the minimal. If there is a tie for the
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hw3sol - CS123 Spring 2010 Solution to Homework 3 Problem 1...

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