UGC NET - Sets and relations: set operations, representation and properties of relations, equivalence relations, partial ordering. Online Test (MCQ Quiz)

1. Which of the following is the correct notation for the union of two sets A and B?

A) A ∩ B B) A ∪ B C) A - B D) A ⊆ B

2. If set A = {1, 2, 3} and set B = {3, 4, 5}, what is the intersection of A and B (A ∩ B)?

A) {1, 2, 3, 4, 5} B) {3} C) {1, 2} D) {4, 5}

3. The difference of set A from set B (B - A) contains elements that are:

A) In A but not in B B) In B and also in A C) In B but not in A D) In either A or B but not both

4. What is the cardinality of the power set of a set with n elements?

A) n B) 2n C) n^2 D) 2^n

5. A relation R on a set A is called reflexive if:

A) For all a ∈ A, (a, a) ∈ R B) For all a, b ∈ A, if (a, b) ∈ R then (b, a) ∈ R C) For all a, b, c ∈ A, if (a, b) ∈ R and (b, c) ∈ R then (a, c) ∈ R D) For all a, b ∈ A, (a, b) ∈ R implies a ≠ b

6. A relation R on a set A is called symmetric if:

A) For all a ∈ A, (a, a) ∈ R B) For all a, b ∈ A, if (a, b) ∈ R then (b, a) ∈ R C) For all a, b, c ∈ A, if (a, b) ∈ R and (b, c) ∈ R then (a, c) ∈ R D) For all a, b ∈ A, (a, b) ∈ R implies a ≠ b

7. A relation R on a set A is called transitive if:

A) For all a ∈ A, (a, a) ∈ R B) For all a, b ∈ A, if (a, b) ∈ R then (b, a) ∈ R C) For all a, b, c ∈ A, if (a, b) ∈ R and (b, c) ∈ R then (a, c) ∈ R D) For all a, b ∈ A, (a, b) ∈ R implies a ≠ b

8. An equivalence relation on a set A is a relation that is:

A) Reflexive only B) Symmetric only C) Transitive only D) Reflexive, symmetric, and transitive

9. If R is an equivalence relation on set A, then the partition of A induced by R consists of:

A) All subsets of A B) The equivalence classes of A under R C) The set A itself D) The empty set

10. A relation R on a set A is a partial ordering if it is:

A) Reflexive, symmetric, and transitive B) Reflexive, antisymmetric, and transitive C) Irreflexive, symmetric, and transitive D) Symmetric, transitive, and total

11. What does it mean for a relation R to be antisymmetric?

A) If (a, b) ∈ R and (b, a) ∈ R, then a = b B) If a = b, then (a, b) ∈ R C) If (a, b) ∈ R, then (b, a) ∉ R D) If (a, b) ∈ R and (b, a) ∈ R, then a ≠ b

12. Consider the set of integers Z. The relation 'divides' (a | b) is:

A) An equivalence relation B) A partial ordering C) Neither an equivalence relation nor a partial ordering D) A total ordering

13. Which of the following is NOT a property of an equivalence relation?

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

14. If A = {1, 2, 3}, which of the following is a reflexive relation on A?

A) {(1, 1), (2, 2), (3, 3), (1, 2)} B) {(1, 1), (2, 3), (3, 2)} C) {(1, 2), (2, 1), (3, 3)} D) {(1, 1), (2, 2), (1, 2)}

15. Let R be the relation 'is less than' (<) on the set of real numbers. Which property does R NOT satisfy?

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

16. The set of all ordered pairs (x, y) such that x is a student and y is a course taken by x is an example of a relation between:

A) Two sets B) A single set C) Three sets D) No sets

17. If A = {a, b} and B = {1, 2}, what is the Cartesian product A × B?

A) {(a, 1), (a, 2), (b, 1), (b, 2)} B) {(1, a), (1, b), (2, a), (2, b)} C) {(a, 1), (b, 2)} D) {(a, b), (1, 2)}

18. Which of the following relations on the set A = {1, 2} is NOT symmetric?

A) {(1, 1), (2, 2)} B) {(1, 2), (2, 1), (1, 1), (2, 2)} C) {(1, 2), (2, 1)} D) {(1, 2), (1, 1), (2, 2)}

19. Let R be the relation 'has the same parity as' on the set of integers. This relation is:

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

20. A relation R on set A is a total ordering if it is a partial ordering and for every a, b ∈ A, either:

A) (a, b) ∈ R or (b, a) ∈ R B) (a, b) ∈ R and (b, a) ∈ R C) (a, b) ∉ R or (b, a) ∉ R D) (a, b) ∉ R and (b, a) ∉ R

21. If R = {(1, 2), (2, 3), (1, 3)} on set A = {1, 2, 3}, what is the transitive closure of R?

A) {(1, 2), (2, 3), (1, 3)} B) {(1, 2), (2, 3), (1, 3), (1, 1), (2, 2), (3, 3)} C) {(1, 2), (2, 3), (1, 3), (1, 3)} D) {(1, 2), (2, 3), (1, 3), (1, 3), (2, 1), (3, 2)}

22. Let A = {1, 2, 3, 4} and R be the relation 'is less than or equal to' (≤). Which of the following is true about R?

A) R is reflexive, symmetric, and transitive. B) R is reflexive, antisymmetric, and transitive. C) R is symmetric and transitive, but not reflexive. D) R is reflexive, but neither symmetric nor transitive.

23. Consider the relation R = {(a, a), (b, b), (c, c), (a, b), (b, a)} on the set S = {a, b, c}. Is R an equivalence relation?

A) Yes, because it is reflexive, symmetric, and transitive. B) No, because it is not reflexive. C) No, because it is not symmetric. D) No, because it is not transitive.

24. What is the complement of a set A, denoted by A', with respect to a universal set U?

A) Elements in A only B) Elements in U but not in A C) Elements in both A and U D) Elements in A or U

25. Let A = {1, 2} and B = {2, 3}. What is A Δ B (symmetric difference)?

A) {2} B) {1, 3} C) {1, 2, 3} D) {1, 2, 2, 3}

26. A relation R on a set A is anti-reflexive if:

A) For all a ∈ A, (a, a) ∈ R B) For all a ∈ A, (a, a) ∉ R C) For all a, b ∈ A, if (a, b) ∈ R then (b, a) ∈ R D) For all a, b ∈ A, if (a, b) ∈ R and (b, c) ∈ R then (a, c) ∈ R

27. The relation 'is a sibling of' on the set of people is:

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

28. If R is a relation on set A, the inverse relation R⁻¹ consists of:

A) All pairs (a, b) such that (a, b) ∈ R B) All pairs (b, a) such that (a, b) ∈ R C) All pairs (a, b) such that (a, b) ∉ R D) All pairs (a, b) such that a = b

29. Which of the following is a property of set union?

A) Commutative only B) Associative only C) Both commutative and associative D) Distributive only

30. Let R be the relation {(1,1), (2,2), (3,3), (1,2), (2,1)} on A = {1,2,3}. R 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

31. The relation 'is a proper subset of' (⊂) on the set of all sets is:

A) Reflexive, symmetric, and transitive B) Reflexive, antisymmetric, and transitive C) Irreflexive, antisymmetric, and transitive D) Symmetric and transitive, but not reflexive

32. If R is an equivalence relation on A, and a ∈ A, then the equivalence class of a, denoted by [a], is:

A) {x ∈ A | (a, x) ∈ R} B) {x ∈ A | (x, a) ∈ R} C) {x ∈ A | (a, x) ∉ R} D) {x ∈ A | x = a}

33. The set of all pairs (x, y) where x and y are integers and x < y, forms a relation that is:

A) Reflexive and symmetric B) Antisymmetric and transitive C) Reflexive and transitive D) Symmetric and transitive

34. Which of the following is the identity relation on set A?

A) {(a, a) | a ∈ A} B) {(a, b) | a, b ∈ A and a ≠ b} C) {(a, b) | a, b ∈ A} D) ∅ (the empty set)

35. If A = {1, 2, 3} and B = {3, 4, 5}, what is A ∪ B?

A) {3} B) {1, 2} C) {1, 2, 3, 4, 5} D) {1, 2, 4, 5}

36. Consider the relation R = {(1,2), (2,1), (1,1)} on set A = {1,2}. R is:

A) Reflexive, symmetric, and transitive B) Reflexive and symmetric, but not transitive C) Symmetric and transitive, but not reflexive D) Reflexive and transitive, but not symmetric

37. The relation 'is a divisor of' on the set of positive integers is:

A) An equivalence relation B) A partial ordering C) A total ordering D) Symmetric but not reflexive

38. If R is a relation on A, and R⁻¹ is its inverse, then R is symmetric if and only if:

A) R = R⁻¹ B) R ∩ R⁻¹ = ∅ C) R ∪ R⁻¹ = A × A D) R⁻¹ ⊆ R

39. Let R be the relation 'has the same number of elements as' on the set of all finite sets. This relation is:

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

40. Which of the following is NOT a property of set intersection?

A) Commutative B) Associative C) Distributive over union D) Idempotent

41. If A = {a, b, c} and R = {(a, a), (b, b), (c, c)}, R is:

A) Reflexive, symmetric, and transitive B) Reflexive and symmetric, but not transitive C) Reflexive and transitive, but not symmetric D) Symmetric and transitive, but not reflexive

42. A relation R on set A is transitive if for all a, b, c ∈ A, if (a, b) ∈ R and (b, c) ∈ R, then:

A) (a, c) ∈ R B) (a, c) ∉ R C) (c, a) ∈ R D) (b, a) ∈ R

43. The set of all pairs (x, y) where x and y are integers and x ≥ y, forms a relation that is:

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

44. If A = {1, 2} and B = {3, 4}, what is A × B?

A) {(1, 3), (1, 4), (2, 3), (2, 4)} B) {(3, 1), (4, 1), (3, 2), (4, 2)} C) {(1, 2), (3, 4)} D) {(1, 3), (2, 4)}

45. Which property is required for a relation to be a partial ordering, but not necessarily for it to be an equivalence relation?

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

46. Let R be the relation 'is married to' on the set of all people. R is:

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

47. If A = {1, 2, 3} and B = {2, 3, 4}, what is A - B?

A) {2, 3} B) {1} C) {4} D) {1, 4}

48. Consider the relation R = {(x, y) | x and y are integers and x - y is even} on the set of integers. R is:

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

49. What is the property that states for any sets A, B, C: A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)?

A) Commutative property of union B) Associative property of intersection C) Distributive property of union over intersection D) Distributive property of intersection over union

50. If A = {1, 2, 3} and R = {(1, 2), (2, 1)}, then the reflexive closure of R is:

A) {(1, 2), (2, 1)} B) {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1)} C) {(1, 1), (2, 2), (1, 2), (2, 1)} D) {(1, 2), (2, 1), (3, 3)}

Sets and relations: set operations, representation and properties of relations, equivalence relations, partial ordering. - Online Test

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