Worksheet on Relations

Transcription

1 Worksheet on Relations Recall the properties that relations can have: Definition. Let R be a relation on the set A. R is reflexive if for all a A we have ara. R is irreflexive or antireflexive if for all a A we have a Ra. R is symmetric if for all a, b A we have arb bra. R is antisymmetric if for all a, b A we have (arb bra) = a = b. R is transitive if for all a, b, c A we have (arb brc) = arc. Example 1. Let A = {1, 2}. (a) List all the elements of A A. (b) List all possible relations we can define on the set A. 1

2 1 Approach A: brute force Example 2. For each R i we can ask 5 questions: Is it reflexive? Is it anti-reflexive? Is it symmetric? Is it anti-symmetric? Is it transitive? This gives us 80 true or false questions. How many do you want to do? Is there a better way to do it? A.1. Is R 1 reflexive? A.21. Is R 5 reflexive? A.41. Is R 9 reflexive? A.61. Is R 13 reflexive? A.2. Is R 1 anti-reflexive? A.22. Is R 5 anti-reflexive? A.42. Is R 9 anti-reflexive? A.62. Is R 13 anti-reflexive? A.3. Is R 1 symmetric? A.23. Is R 5 symmetric? A.43. Is R 9 symmetric? A.63. Is R 13 symmetric? A.4. Is R 1 anti-symmetric? A.24. Is R 5 anti-symmetric? A.44. Is R 9 anti-symmetric? A.64. Is R 13 anti-symmetric? A.5. Is R 1 transitive? A.25. Is R 5 transitive? A.45. Is R 9 transitive? A.65. Is R 13 transitive? A.6. Is R 2 reflexive? A.26. Is R 6 reflexive? A.46. Is R 10 reflexive? A.66. Is R 14 reflexive? A.7. Is R 2 anti-reflexive? A.27. Is R 6 anti-reflexive? A.47. Is R 10 anti-reflexive? A.67. Is R 14 anti-reflexive? A.8. Is R 2 symmetric? A.28. Is R 6 symmetric? A.48. Is R 10 symmetric? A.68. Is R 14 symmetric? A.9. Is R 2 anti-symmetric? A.29. Is R 6 anti-symmetric? A.49. Is R 10 anti-symmetric? A.69. Is R 14 anti-symmetric? A.10. Is R 2 transitive? A.30. Is R 6 transitive? A.50. Is R 10 transitive? A.70. Is R 14 transitive? A.11. Is R 3 reflexive? A.31. Is R 7 reflexive? A.51. Is R 11 reflexive? A.71. Is R 15 reflexive? A.12. Is R 3 anti-reflexive? A.32. Is R 7 anti-reflexive? A.52. Is R 11 anti-reflexive? A.72. Is R 15 anti-reflexive? A.13. Is R 3 symmetric? A.33. Is R 7 symmetric? A.53. Is R 11 symmetric? A.73. Is R 15 symmetric? A.14. Is R 3 anti-symmetric? A.34. Is R 7 anti-symmetric? A.54. Is R 11 anti-symmetric? A.74. Is R 15 anti-symmetric? A.15. Is R 3 transitive? A.35. Is R 7 transitive? A.55. Is R 11 transitive? A.75. Is R 15 transitive? A.16. Is R 4 reflexive? A.36. Is R 8 reflexive? A.56. Is R 12 reflexive? A.76. Is R 16 reflexive? A.17. Is R 4 anti-reflexive? A.37. Is R 8 anti-reflexive? A.57. Is R 12 anti-reflexive? A.77. Is R 16 anti-reflexive? A.18. Is R 4 symmetric? A.38. Is R 8 symmetric? A.58. Is R 12 symmetric? A.78. Is R 16 symmetric? A.19. Is R 4 anti-symmetric? A.39. Is R 8 anti-symmetric? A.59. Is R 12 anti-symmetric? A.79. Is R 16 anti-symmetric? A.20. Is R 4 transitive? A.40. Is R 8 transitive? A.60. Is R 12 transitive? A.80. Is R 16 transitive?

3 Example 3. The relations and properties can be arranged in a grid like this. How many would you like to mark as T/F? Is there more we can do to reveal the patterns? Reflexive antireflexive symmetric antisymmetric transitive R 1 R 2 R 3 R 4 R 5 R 6 R 7 R 8 R 9 R 10 R 11 R 12 R 13 R 14 R 15 R 16 3

4 2 Approach B: Logic and Patterns As you, a young mathematician, contemplate answering all the questions about which relation has which property, you should naturally start wondering if there are patterns, shortcuts, etc. There are definitely some logical connections you can make and let me lay a little groundwork for them. 2.1 Inheritance of Properties Let s turn the problem around: once we ve shown that a certain relation has a certain property, maybe that property is inherited by other relations. Here s what I mean by that. Example 4. (a) Show that the reflexive property is inherited by supersets (i.e. if R is reflexive and R S then S is reflexive). (b) Show that the antireflexive property is inherited by subsets. (i.e. if R is antireflexive and S R then S is antireflexive). (c) Show that the symmetric property is inherited by unions and intersections (i.e. if R and S are symmetric then so is R S and R S). (d) Show that the antisymmetric property is inherited by subsets. (i.e. if R is antisymmetric and S R then S is antisymmetric). (e) Show that the transitive property is inherited by intersections. (i.e. if R and S are transitive then so is R S). Example 5. Fill in the indicated properties that are inherited, or not, in the table below T/F for inherited Reflexive antireflexive symmetric antisymmetric transitive supersets subsets intersections unions 4

5 2.2 Hasse Diagrams To see how the above concepts might be helpful, let s make a Hasse diagram of the relations. If A is a finite set that has a partial order, we often picture the elements of A and their order using what s called a Hasse diagram. A Hasse diagram is a type of directed graph where we represent elements of A in a picture. Given two elements x and y with xry, and where no element is between them in the partial order, then we place y higher up in the picture and connect them with an edge. For instance, here are two examples of Hasse diagrams A = {1, 2, 3, 4} and is the usual order. 1 A = {x N x 12} and is the order. For instance in the second diagram we see that 2 4 and that We don t show, but it is implied by transitivity, that We see that both 2 and 3 divide 6, etc. You can see also least common multiples, e.g. to find the least common multiple of 4 and 6 go up until you find the smallest element that is connected to both of them. To find the greatest common divisor you go down until you find the largest element that is connected to both numbers. Example 6. Some of our relations are partial orders, but the collection of all our relations is a partial order. The order is. Make the Hasse diagram for our relations R 1,..., R 16. (Remember, relations are just subsets of A A, so this is really the Hasse diagram for P(A A). If you like you can cheat and look at the Wikipedia page on Hasse diagrams. But remember to use our notation, R 1,..., R 16. For instance, you should have R 2 R 6 in the diagram, with R 6 higher up the diagram.) 5

6 Before you do the next three problems, you might want to do two things. (1) Make three copies of your Hasse diagram from the previous problem. (2) Practice looking at Hasse diagrams to see which set contains which one, and also, therefore, to see intersections and unions. So, for instance, you should be able to look at your diagram to see that R 3 R 10, because there should be a line going up from R 3 directly to R 10. In a similar way, consider R 9 and R 11. Neither of these contains the other, since there is no line connecting them. But, you should be able to see that R 9 R 11 = R 15, because there is a line going up from R 9 to R 15, and a line going up from R 11 to R 15. So R 15 contains both R 9 and R 11, and it is the smallest set that contains them both. In a similar way you should be able find the union of any two sets. Similarly, you should be able to see that R 9 R 11 = R 4, since there are lines going down from R 9 and R 11 directly to R 4. In a similar way you should be able to find the intersection of any two sets. As you do the next three problems, always be looking for patterns in the diagrams based on subsets, supersets, intersections and unions. Example 7. Take your first copy of the Hasse diagram and pick two colors, one for reflexive and one for antireflexive. Color R 8 as reflexive. The reflexive property is inherited by supersets. Use this to color the rest of the nodes on the diagram that are reflexive. Color R 9 as antireflexive. The antireflexive property is inherited by subsets. Use this to color the rest of the nodes on the diagram that are reflexive. Example 8. Take another copy of your diagram, and use two colors: one for symmetric and one for antisymmetric. Color in R 2, R 5, R 12 and R 15 for being symmetric. The property of symmetric is inherited by unions and by intersections. Apply these two facts to color the rest of the relations that are symmetric. Color in R 13 and R 14 for being antisymmetric. The property of antisymmetric is inherited by subsets. Use this to color the rest of the relations that are antisymmetric. Example 9. Take another copy of your diagram, and use one color for transitive. Color in R 6, R 7, R 10, R 11, R 13, R 14 and R 16 for being transitive. The transitive property is inherited by intersections. Use this fact to color in the rest of the relations that are transitive. 6

7 2.3 Overlapping Properties for Relations Let X be any nonempty set. Define two particular, single relations on X = the empty relation (i.e. nothing is related to anything else E = equality = {(x, y) X X x = y} Example 10. The picture below is a Venn diagram on the collection of relations (based on this article by Pfeiffer): Reflexive Transitive Antireflexive E Antisymmetric Symmetric The diagram is meant to be interpreted as follows: The two small black circles represent unique relations. The black circle on the left is E: the only relation that is reflexive, transitive, symmetric and antisymmetric. The black circle on the right is the empty relation. The intersections of different collections are shown as is usual in a Venn diagram. For example, relations that are both reflexive and transitive would be shown in the overlapping ovals labeled with those properties. The black shaded area shows combinations that do not exist. See how many logical statements about relations you can come up with based on the diagram above. Extra Credit: (1) Can you redraw the above Venn diagram in some way to make it nicer? Be creative, come up with your own ideas for making it nicer. (Here are some prompts: use colors. Eliminate areas that are filled in since they represent combinations that don t exist. Maybe represent subsets of E somehow.) (2) How many properties can you write down based on the above Venn diagram? (3) Prove some of the properties you just wrote down. 2.4 Counting relations Example 11. How many relations out of R 1,...,R 16 are reflexive? Antireflexive? Symmetric? Antisymmetric? Transitive? Partial orders? Equivalence relations? 7