Sets and relations: set operations, representation and properties of relations, equivalence relations, partial ordering. - Question Bank

1. 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)}
2. 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
3. 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
4. If A = {1, 2, 3} and B = {2, 3, 4}, what is A - B?
A) {2, 3}
B) {1}
C) {4}
D) {1, 4}
5. 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
6. 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
7. 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)}
8. 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
9. 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
10. 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
11. Which of the following is NOT a property of set intersection?
A) Commutative
B) Associative
C) Distributive over union
D) Idempotent
12. 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
13. 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
14. 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
15. 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
16. 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}
17. 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)
18. 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
19. 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}
20. 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
21. 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
22. Which of the following is a property of set union?
A) Commutative only
B) Associative only
C) Both commutative and associative
D) Distributive only
23. 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
24. 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
25. 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
26. 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}
27. 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
28. 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.
29. 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.
30. 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)}
31. 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
32. 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
33. 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)}
34. 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)}
35. 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
36. 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
37. 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)}
38. Which of the following is NOT a property of an equivalence relation?
A) Reflexivity
B) Symmetry
C) Antisymmetry
D) Transitivity
39. 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
40. 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
41. 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
42. 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
43. 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
44. 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
45. 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
46. 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
47. What is the cardinality of the power set of a set with n elements?
A) n
B) 2n
C) n^2
D) 2^n
48. 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
49. 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}
50. 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
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