Please answer item 27 only 3.4 ExercisesIn Exercises 1-10, determine whether the given relation is anequivalence relation on {1, 2, 3, 4, 5). If the relation is an equiv-alence relation, list the equivalence classes. (In Exercises 5-10,x, ye (1, 2, 3, 4, 5).)1. ((1, 1), (2, 2), (3, 3). (4,4). (5,5). (1,3), (3, 1))2. {(1, 1), (2, 2), (3, 3), (4,4), (5,5), (1, 3), (3, 1), (3, 4). (4,3)}3. ((1, 1), (2, 2), (3, 3), (4,4)}4. {(1, 1), (2, 2), (3, 3), (4,4), (5, 5), (1, 5), (5. 1), (3, 5), (5,3).(1,3), (3, 1))5. ((x,y) | 1 ≤x≤ 5 and 1 ≤ y ≤ 5}6. ((x, y) | 4 divides .x - y)8. {(x, y) |x divides 2-y}7. {(x, y) | 3 divides x + y}9. {(x, y) |x and y are both even}10. {(x, y) |x and y are both even or x and y are both odd}In Exercises 11-16, determine whether the given relation is anequivalence relation on the set of all people.11. {(x, y) |x and y are the same height}12. {(x, y) [x and y have, at some time, lived in the same country}13. {(x, y) |x and y have the same first name}14. ((x, y) |x is taller than y}15. ((x, y) [x and y have the same parents}16. ((x, y) [x and y have the same color hair}In Exercises 17-22, list the members of the equivalence relationon (1,2,3,4) defined (as in Theorem 3.4.1) by the given partition.Also, find the equivalence classes [1], [2], [3], and [4].17. ((1.2), (3, 4))19. (1), (2), (3), (4))21. (1,2,3,4}}18. (1), (2), (3,4))20. (1, 2, 3), (4}}22.(1), (2, 4), (3}}In Exercises 23-25, let X=(1, 2, 3, 4, 5), Y=(3, 4), and C=(1.3). Define the relation R on P(X), the set of all subsets of X.asARBifAUY=BUY.23. Show that R is an equivalence relation.24. List the elements of [C], the equivalence class containing C.25. How many distinct equivalence classes are there?26. LetX = (San Francisco, Pittsburgh, Chicago, San Diego,Philadelphia, Los Angeles).Define a relation R on X as x Ry if x and y are in the same state.(a) Show that R is an equivalence relation.(b) List the equivalence classes of X.27. If an equivalence relation has only one equivalence class, whatmust the relation look like?28. If R is an equivalence relation on a finite set X and [X] = [R].what must the relation look like?29. By listing ordered pairs, give an example of an equivalencerelation on [1, 2, 3, 4, 5, 6) having exactly four equivalenceclasses.30. How many equivalence relations are there on the set (1, 2, 3)?31. Let R be a reflexive relation on X satisfying: for all x, y, z € X,if x Ry and y Rz, then z R.x. Prove that R is an equivalence re-lation.32. Define a relation R on RR, the set of functions from R to R.byf Rg if f(0) = g(0). Prove that R is an equivalence relationon RR. Let f(x) = x for all x € R. Describe [/].33. Define a relation R on RR, the set of functions from R to R byfRg if there exist a, b R such that f(x) = g(x + a) + bfor all x € R. Prove that R is an equivalence relation onRR. What property do all functions in an equivalence classshare?34. Let X = [1,2,.... 10). Define a relation R on X X X by(a, b) R (c,d) if a+d=b+c.(a) Show that R is an equivalence relation on X X X.(b) List one member of each equivalence class of X X X.35. Let X = (1,2,.... 10). Define a relation R on X X X by(a, b) R (c,d) if ad=bc.(a) Show that R is an equivalence relation on X X X.(b) List one member of each equivalence class of X X X.(c) Describe the relation R in familiar terms.36. Let R be a reflexive and transitive relation on X. Show thatROR is an equivalence relation on X.

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27. If an equivalence relation has only one equivalence class, what must the relation look like?