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Solutions to Tutorial Exercises: Week 6 Logik für Linguisten: Universität Leipzig, Nov. 27 - Dec. 4, 2014 Sandhya Sundaresan 1. A = {red, blue, green, yellow}. R1 , R2 , R3 , and R4 are relations defined on A. • R1 = {<red, red>, <red, blue>, <green, green>, <yellow, green>, <red, green>, <blue, blue>, <blue, green>, <yellow, yellow>}. • R2 = {<red, blue>, <blue, yellow>, <red, yellow>, <green, green>, <blue, red>} • R3 = {<red, yellow>, <green, blue>, <blue, red>, <green, red>, <green, yellow>, <yellow, blue>} • R4 = {<red, red>, <green, green>, <blue, blue>, <yellow, yellow>} • R5 = {<yellow, yellow>, <blue, green>, <yellow, red>, <blue, yellow>, <blue, red>, <red, red>, <green, red>, <green, green>, <green, yellow>, <blue, blue>} Now answer the following questions: i. Represent the relations above in terms of diagrams (with arrows, as shown in class). Answer: Students should do these as shown in class, with arrows going from the first element of every ordered pair in the relation to the second. ii. Describe the properties of the relations, their inverses, and complements above with respect to: reflexivity, symmetry, transitivity, and connectedness. Answer: The relations R1 – R5 , their inverses and their complements, have the following properties: • R1 and its inverse R1 ´1 are both: transitive, reflexive, antisymmetric, and nonconnected. The complement R1 ’ is: irreflexive, non-symmetric, non-transitive, and connected. • R2 and R2 ´1 are both: non-reflexive, non-symmetric, transitive, and non-connected. R2 ’ is: non-reflexive, non-symmetric, non-transitive, and non-connected. • R3 and R3 ´1 are both: irreflexive, asymmetric, non-transitive, and connected. R3 ’ is: reflexive, antisymmetric, non-transitive, and connected. • R4 and R4 ´1 are both: reflexive, antisymmetric, transitive, and non-connected (note that R4 = R4 ´1 . R4 ’ is: irreflexive, symmetric, non-transitive, and connected. • R5 and R5 ´1 are both: reflexive, antisymmetric, transitive, and connected. R5 ’ is: irreflexive, asymmetric, transitive, and connected. iii. Which of the relations R1 – R5 represents an order? And are these orders weak or strong? Explain. Answer: R1 , R4 , and R5 all represent orders. They are all weak orders. This is because they are all reflexive, transitive, and antisymmetric, which is the definition of a weak order. iv. For each of the weak orders in R1 – R5 above, give the corresponding strong order. For each of the strong orders above, give the corresponding weak order. Answer: The following strong orders may be derived from the weak ones: • The strong order corresponding to R1 is: {<red, blue>, <yellow, green>, <red, green>, <blue, green>}. Note that it is: irreflexive, asymmetric, and transitive, just as desired. 1 Sandhya Sundaresan Lecture 6 • The strong order corresponding to R4 (let’s call this strong order S4 ) is the H. It makes sense to think of H as a strong order for the following reasons: it is irreflexive because, of course, it is not the case that every element in the set A of which S4 is a relation is represented in S4 in the form <x, x>; in fact, none of the elements in A are represented in this manner in S4 . S4 is transitive: transitivity requires that if there is an ordered pair <x, y> and another ordered pair <y, z> in S4 , then there should also be an ordered pair <x, z> in S4 . But S4 doesn’t have any two ordered pairs <x, y> and <y, z> (since it has no members at all!), so the condition is trivially satisfied. Similarly, S4 is also asymmetric: asymmetry requires that there can be no ordered pairs <x, y> and <y, x> in S4 and indeed this is satisfied, since S4 has no ordered pairs at all. (In fact, by similar reasoning, the empty set is also symmetric!). • The strong order corresponding to R5 is: {<blue, green>, <yellow, red>, <blue, yellow>, <blue, red>, <green, red>, <green, yellow>}. Note that it is: irreflexive, asymmetric, and transitive, just as desired. v. Which of the orders above represents a total/linear order? Explain. Answer: Of the weak orders in R1 , R4 and R5 , only R5 is connected. Thus, only R5 represents a total/linear order. vi. What are the least, greatest, maximal, and minimal elements of the orders above? Explain. R1 : The elements red and yellow are both minimal; green is maximal and greatest in R1 . There is no least element. R4 : The elements red, green, blue, yellow are both maximal and minimal in R4 . There is no least and greatest element. R5 : red is both maximal and greatest. blue is both minimal and least. vii. Which elements in the orders immediately precede other ones? Answer: red immediately precedes blue which immediately precedes green, and yellow immediately precedes green in R1 . In R4 , no elements immediately precede each other (since the definition of immediate precedence is defined over distinct elements x and y). In R5 : yellow immediately precedes red, and green immediately precedes yellow, and blue immediately precedes green. 2. R is a set of ordered pairs ă a, b ą defined on the set of human beings such that the shoe-size of the right foot of a is less than the shoe-size of the right foot of b (assuming the Continental European system of shoe-sizes: same scale for men, women, and children). Now say which of the following properties describe R and explain your answer: • Weak order: Answer: In order to be a weak order, the relation has to be: reflexive, transitive, and antisymmetric. We can immediately see that the relation above is not a weak order because it is irreflexive: a human being x can never have a smaller right shoe size than themselves. • Strong order Answer: In order to be a strong order, the relation has to be: irreflexive, transitive, and asymmetric. This is satisfied: the relation is irreflexive (as explained above); it is also transitive: if x has a smaller shoe-size than y in the right foot, and y has a smaller shoe-size than z in the right foot, then x necessarily has a smaller shoe-size than z in 2 Logik für Linguisten Nov. 27 - Dec. 4, 2014 the right foot. Finally, the relation is also asymmetric: we will never find a pair <x, y> in the relation above for which there is an inverse pair <y, x>. • An order having a maximal, minimal, greater or least element Answer: Yes, R does have a maximal element or elements: this will denote the human being or human beings who have a larger shoe size than anyone else. Similarly R also has a minimal element/elements: this will denote the human being or human beings who have a smaller shoe-size than anyone else. However, R need not have a greatest or least element: after all, it need not be the case that there is a unique individual in the world who has a smaller shoe size than everyone else; similarly, it also need not be the case that there is a unique individual in the world who has a larger shoe-size than anyone else. • A total/linear order: No, R is not a total/linear order. After all, two distinct human beings who have the same shoe size will not be a part of the relation above. Thus, R is not a connected relation. • Defined on a well-ordered set: Since R is not a total order, it is also not defined on a well-ordered set (which requires that the order be total). • A dense order: No, R is not a dense order. Shoe-sizes are discrete: thus there will be no distinct shoe-size between two consecutive shoe-sizes. 3