JEE Main - Equivalence relations Online Test (MCQ Quiz)

1. Which property must a relation R on a set A satisfy to be considered an equivalence relation?

A) Reflexive and Symmetric B) Reflexive and Transitive C) Symmetric and Transitive D) Reflexive, Symmetric, and Transitive

2. A relation R on a set A is reflexive if:

A) For every a ∈ A, (a, a) ∈ R B) For every a, b ∈ A, if (a, b) ∈ R, then (b, a) ∈ R C) For every a, b, c ∈ A, if (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R D) For every a ∈ A, there exists b ∈ A such that (a, b) ∈ R

3. A relation R on a set A is symmetric if:

A) For every a ∈ A, (a, a) ∈ R B) For every a, b ∈ A, if (a, b) ∈ R, then (b, a) ∈ R C) For every a, b, c ∈ A, if (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R D) For every a ∈ A, there exists b ∈ A such that (a, b) ∈ R

4. A relation R on a set A is transitive if:

A) For every a ∈ A, (a, a) ∈ R B) For every a, b ∈ A, if (a, b) ∈ R, then (b, a) ∈ R C) For every a, b, c ∈ A, if (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R D) For every a ∈ A, there exists b ∈ A such that (a, b) ∈ R

5. Consider the set A = {1, 2, 3} and the relation R = {(1, 1), (2, 2), (3, 3)}. Is R an equivalence relation?

A) No, it is not reflexive. B) No, it is not symmetric. C) No, it is not transitive. D) Yes, it is an equivalence relation.

6. 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?

A) No, it is not reflexive. B) No, it is not transitive. C) Yes, it is an equivalence relation. D) No, it is not symmetric.

7. 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?

A) No, it is not reflexive. B) No, it is not symmetric. C) No, it is not transitive. D) Yes, it is an equivalence relation.

8. 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?

A) Reflexive B) Symmetric C) Transitive D) None, R is an equivalence relation.

9. 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?

A) Reflexive B) Symmetric C) Transitive D) None, R is an equivalence relation.

10. 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?

A) Reflexive B) Symmetric C) Transitive D) None, R is an equivalence relation.

11. The relation 'is parallel to' on the set of all lines in a plane is:

A) Reflexive only B) Symmetric only C) Transitive only D) An equivalence relation

12. The relation 'is perpendicular to' on the set of all lines in a plane is:

A) Reflexive B) Symmetric C) Transitive D) An equivalence relation

13. 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?

A) Reflexive B) Symmetric C) Transitive D) None of these

14. 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?

A) No, it is not reflexive. B) No, it is not symmetric. C) No, it is not transitive. D) Yes, it is an equivalence relation.

15. 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?

A) No, it is not reflexive. B) No, it is not symmetric. C) No, it is not transitive. D) Yes, it is an equivalence relation.

16. 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?

A) No, it is not reflexive. B) No, it is not symmetric. C) No, it is not transitive. D) Yes, it is an equivalence relation.

17. Let A be a non-empty set. The empty relation {} on A is:

A) An equivalence relation B) Reflexive and symmetric C) Transitive but not reflexive or symmetric D) Neither reflexive, symmetric, nor transitive

18. Let A be a non-empty set. The universal relation A x A on A is:

A) Reflexive and symmetric but not transitive B) Reflexive and transitive but not symmetric C) Symmetric and transitive but not reflexive D) An equivalence relation

19. If R is an equivalence relation on a set A, then the equivalence class of an element 'a' is denoted by:

A) [a] B) {a} C) aR D) E(a)

20. The set of all equivalence classes of an equivalence relation R on a set A forms a:

A) Union of disjoint sets B) Partition of the set A C) Subset of A D) Superset of A

21. 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?

A) No, it is not reflexive. B) No, it is not symmetric. C) No, it is not transitive. D) Yes, it is an equivalence relation.

22. 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?

A) No, it is not reflexive. B) No, it is not symmetric. C) No, it is not transitive. D) Yes, it is an equivalence relation.

23. 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?

A) No, it is not reflexive. B) No, it is not symmetric. C) No, it is not transitive. D) Yes, it is an equivalence relation.

24. 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?

A) No, it is not reflexive. B) No, it is not symmetric. C) No, it is not transitive. D) Yes, it is an equivalence relation.

25. 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?

A) No, it is not reflexive. B) No, it is not symmetric. C) No, it is not transitive. D) Yes, it is an equivalence relation.

26. 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?

A) No, it is not reflexive. B) No, it is not symmetric. C) No, it is not transitive. D) Yes, it is an equivalence relation.

27. 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?

A) Add (2, 1) B) Add (1, 3) C) Add (2, 1) and (2, 3) D) Add (2, 1) and (3, 1)

28. 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?

A) Add (1, 3) B) Add (3, 1) C) Add (1, 3) and (3, 1) D) No property needs to be added.

29. If a relation R is reflexive and symmetric, it is necessarily:

A) Transitive B) An equivalence relation C) Not transitive D) None of the above

30. If a relation R is symmetric and transitive, it is necessarily:

A) Reflexive B) An equivalence relation C) Not reflexive D) None of the above

31. If a relation R is reflexive and transitive, it is necessarily:

A) Symmetric B) An equivalence relation C) Not symmetric D) None of the above

32. Let R be an equivalence relation on a set A. If a R b, then the equivalence class of a, [a], is equal to:

A) [b] B) {a} C) A D) {a, b}

33. 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?

A) No, it is not reflexive. B) No, it is not symmetric. C) No, it is not transitive. D) Yes, it is an equivalence relation.

34. 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?

A) No, it is not reflexive. B) No, it is not symmetric. C) No, it is not transitive. D) Yes, it is an equivalence relation.

35. 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?

A) No, it is not reflexive. B) No, it is not symmetric. C) No, it is not transitive. D) Yes, it is an equivalence relation.

36. 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?

A) No, it is not reflexive. B) No, it is not symmetric. C) No, it is not transitive. D) Yes, it is an equivalence relation.

37. Let A = {1, 2, 3}. Which of the following relations is an equivalence relation?

A) R1 = {(1, 1), (2, 2), (3, 3), (1, 2)} B) R2 = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1)} C) R3 = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (1, 3)} D) R4 = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (1, 3), (3, 1)}

38. 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?

A) No, it is not reflexive. B) No, it is not symmetric. C) No, it is not transitive. D) Yes, it is an equivalence relation.

39. If R is an equivalence relation on A, and a ∈ A, then a belongs to which equivalence class?

A) The equivalence class of any element b such that a R b B) The equivalence class of any element b such that a is not related to b C) Only the equivalence class of a D) No equivalence class

40. 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?

A) No, it is not reflexive. B) No, it is not symmetric. C) No, it is not transitive. D) Yes, it is an equivalence relation.

41. 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?

A) No, it is not reflexive. B) No, it is not symmetric. C) No, it is not transitive. D) Yes, it is an equivalence relation.

42. 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:

A) The original set B) The empty set C) A subset of the original set D) Undefined

43. 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?

A) No, it is not reflexive. B) No, it is not symmetric. C) No, it is not transitive. D) Yes, it is an equivalence relation.

44. 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?

A) No, it is not reflexive. B) No, it is not symmetric. C) No, it is not transitive. D) Yes, it is an equivalence relation.

45. Let R be a relation on set A. If R is reflexive, symmetric, and transitive, then R is called:

A) A partial order relation B) An equivalence relation C) A strict order relation D) A non-reflexive relation

46. Which of the following is NOT a property required for a relation to be an equivalence relation?

A) Reflexivity B) Symmetry C) Transitivity D) Irreflexivity

47. 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?

A) No, it is not reflexive. B) No, it is not symmetric. C) No, it is not transitive. D) Yes, it is an equivalence relation.

48. 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?

A) No, it is not reflexive. B) No, it is not symmetric. C) No, it is not transitive. D) Yes, it is an equivalence relation.

49. 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?

A) No, it is not reflexive. B) No, it is not symmetric. C) No, it is not transitive. D) Yes, it is an equivalence relation.

50. The relation 'is a factor of' on the set of positive integers is:

A) An equivalence relation B) Reflexive and transitive but not symmetric C) Symmetric and transitive but not reflexive D) Reflexive and symmetric but not transitive

Equivalence relations - Online Test

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