Equivalence relations - Question Bank
1. The relation 'is a factor of' on the set of positive integers is:
2. Let R be the relation on the set of all people, where a R b if a and b have the same number of children. Is R an equivalence relation?
3. Let R be the relation on the set of all people, where a R b if a and b are born in the same month. Is R an equivalence relation?
4. Let R be the relation on the set of all people, where a R b if a and b are siblings. Is R an equivalence relation?
5. Which of the following is NOT a property required for a relation to be an equivalence relation?
6. Let R be a relation on set A. If R is reflexive, symmetric, and transitive, then R is called:
7. Consider the relation R on the set of all subsets of a given set S, where A R B if A ⊆ B. Is R an equivalence relation?
8. Let A = {1, 2, 3, 4, 5, 6} and R be the relation a R b if a and b have the same remainder when divided by 3. Is R an equivalence relation?
9. In the context of equivalence relations, the equivalence classes partition the set. This means that the union of all equivalence classes is the original set, and the intersection of any two distinct equivalence classes is:
10. Let R be the relation on the set of pairs of integers (x, y) such that x*y = x'*y' if (x, y) R (x', y'). Is R an equivalence relation?
11. Let R be the relation on the set of pairs of integers (x, y) such that x+y = x'+y' if (x, y) R (x', y'). Is R an equivalence relation?
12. If R is an equivalence relation on A, and a ∈ A, then a belongs to which equivalence class?
13. Let A = {1, 2, 3, 4}. The relation R = {(a, b) | a and b are both even or a and b are both odd}. Is R an equivalence relation?
14. Let A = {1, 2, 3}. Which of the following relations is an equivalence relation?
15. Consider the relation R on the set of all strings of 'a's and 'b's, where s1 R s2 if s1 and s2 have the same number of 'a's. Is R an equivalence relation?
16. Let R be the relation on the set of rational numbers Q defined by a R b if a - b is an integer. Is R an equivalence relation?
17. Let R be the relation on the set of positive integers N defined by a R b if a = b. Is R an equivalence relation?
18. Let R be the relation on the set of positive integers N defined by a R b if a divides b. Is R an equivalence relation?
19. Let R be an equivalence relation on a set A. If a R b, then the equivalence class of a, [a], is equal to:
20. If a relation R is reflexive and transitive, it is necessarily:
21. If a relation R is symmetric and transitive, it is necessarily:
22. If a relation R is reflexive and symmetric, it is necessarily:
23. Let A = {1, 2, 3} and R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1)}. Which property must be added to R to make it an equivalence relation?
24. Let A = {1, 2, 3} and R = {(1, 1), (2, 2), (3, 3), (1, 2)}. Which property must be added to R to make it an equivalence relation?
25. Let R be the relation on the set of employees in a company, where a R b if a and b are in the same department. Is R an equivalence relation?
26. Consider the relation R on the set of triangles in a plane, where T1 R T2 if T1 is congruent to T2. Is R an equivalence relation?
27. Consider the relation R on the set of triangles in a plane, where T1 R T2 if T1 is similar to T2. Is R an equivalence relation?
28. Let R be the relation on the set of points in a plane defined by (x1, y1) R (x2, y2) if x1 + y1 = x2 + y2. Is R an equivalence relation?
29. Let R be the relation on the set of points in a plane defined by (x1, y1) R (x2, y2) if y1 = y2. Is R an equivalence relation?
30. Let R be the relation on the set of points in a plane defined by (x1, y1) R (x2, y2) if x1 = x2. Is R an equivalence relation?
31. The set of all equivalence classes of an equivalence relation R on a set A forms a:
32. If R is an equivalence relation on a set A, then the equivalence class of an element 'a' is denoted by:
33. Let A be a non-empty set. The universal relation A x A on A is:
34. Let A be a non-empty set. The empty relation {} on A is:
35. Consider the relation R on the set of real numbers R, where x R y if x - y is an integer. Is R an equivalence relation?
36. Let R be the relation on the set of integers Z defined by a R b if a - b is an even number. Is R an equivalence relation?
37. Let R be the relation on the set of integers Z defined by a R b if a + b is even. Is R an equivalence relation?
38. Consider the relation R on the set of integers Z, where a R b if and only if a - b is divisible by 5. Which property does R NOT possess?
39. The relation 'is perpendicular to' on the set of all lines in a plane is:
40. The relation 'is parallel to' on the set of all lines in a plane is:
41. Let A = {a, b, c} and R = {(a, a), (b, b), (c, c), (a, b), (b, a), (a, c), (c, a)}. Which property is missing for R to be an equivalence relation?
42. Let A = {a, b, c} and R = {(a, a), (b, b), (c, c), (a, b), (b, a)}. Which property is missing for R to be an equivalence relation?
43. Let A = {1, 2, 3, 4} and R be the relation 'is equal to' on A. R = {(1, 1), (2, 2), (3, 3), (4, 4)}. Which property does R lack to be an equivalence relation?
44. Consider the set A = {1, 2, 3} and the relation R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (1, 3), (3, 1)}. Is R an equivalence relation?
45. Consider the set A = {1, 2, 3} and the relation R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1)}. Is R an equivalence relation?
46. Consider the set A = {1, 2, 3} and the relation R = {(1, 1), (2, 2), (3, 3)}. Is R an equivalence relation?
47. A relation R on a set A is transitive if:
48. A relation R on a set A is symmetric if:
49. A relation R on a set A is reflexive if:
50. Which property must a relation R on a set A satisfy to be considered an equivalence relation?