Sets and relations: set operations, representation and properties of relations, equivalence relations, partial ordering. - One Line Questions

1. A relation R on set A is a total ordering if it is a partial ordering and for every a, b ∈ A, either: — (a, b) ∈ R or (b, a) ∈ R
2. 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, c) ∈ R
3. If A = {1, 2, 3}, which of the following is a reflexive relation on A? — {(1, 1), (2, 2), (3, 3), (1, 2)}
4. Which of the following relations on the set A = {1, 2} is NOT symmetric? — {(1, 2), (1, 1), (2, 2)}
5. If A = {1, 2, 3} and R = {(1, 2), (2, 1)}, then the reflexive closure of R is: — {(1, 1), (2, 2), (1, 2), (2, 1)}
6. If R = {(1, 2), (2, 3), (1, 3)} on set A = {1, 2, 3}, what is the transitive closure of R? — {(1, 2), (2, 3), (1, 3)}
7. If A = {1, 2} and B = {3, 4}, what is A × B? — {(1, 3), (1, 4), (2, 3), (2, 4)}
8. If A = {a, b} and B = {1, 2}, what is the Cartesian product A × B? — {(a, 1), (a, 2), (b, 1), (b, 2)}
9. Which of the following is the identity relation on set A? — {(a, a) | a ∈ A}
10. If set A = {1, 2, 3} and set B = {3, 4, 5}, what is the intersection of A and B (A ∩ B)? — {3}
11. If A = {1, 2, 3} and B = {2, 3, 4}, what is A - B? — {1}
12. Let A = {1, 2} and B = {2, 3}. What is A Δ B (symmetric difference)? — {1, 3}
13. If A = {1, 2, 3} and B = {3, 4, 5}, what is A ∪ B? — {1, 2, 3, 4, 5}
14. If R is an equivalence relation on A, and a ∈ A, then the equivalence class of a, denoted by [a], is: — {x ∈ A | (a, x) ∈ R}
15. Which of the following is the correct notation for the union of two sets A and B? — A ∪ B
16. If R is a relation on set A, the inverse relation R⁻¹ consists of: — All pairs (b, a) such that (a, b) ∈ R
17. If R is an equivalence relation on set A, then the partition of A induced by R consists of: — The equivalence classes of A under R
18. Consider the set of integers Z. The relation 'divides' (a | b) is: — A partial ordering
19. The relation 'is a divisor of' on the set of positive integers is: — A partial ordering
20. Which of the following is NOT a property of set intersection? — Idempotent
21. Which of the following is a property of set union? — Both commutative and associative
22. What is the property that states for any sets A, B, C: A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)? — Distributive property of union over intersection
23. What is the complement of a set A, denoted by A', with respect to a universal set U? — Elements in U but not in A
24. A relation R on a set A is called reflexive if: — For all a ∈ A, (a, a) ∈ R
25. A relation R on a set A is called symmetric if: — For all a, b ∈ A, if (a, b) ∈ R then (b, a) ∈ R
26. A relation R on a set A is called transitive if: — For all a, b, c ∈ A, if (a, b) ∈ R and (b, c) ∈ R then (a, c) ∈ R
27. A relation R on a set A is anti-reflexive if: — For all a ∈ A, (a, a) ∉ R
28. What does it mean for a relation R to be antisymmetric? — If (a, b) ∈ R and (b, a) ∈ R, then a = b
29. The difference of set A from set B (B - A) contains elements that are: — In B but not in A
30. What is the cardinality of the power set of a set with n elements? — 2^n
31. If R is a relation on A, and R⁻¹ is its inverse, then R is symmetric if and only if: — R = R⁻¹
32. 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? — R is reflexive, antisymmetric, and transitive.
33. The set of all pairs (x, y) where x and y are integers and x < y, forms a relation that is: — Antisymmetric and transitive
34. Let R be the relation 'has the same parity as' on the set of integers. This relation is: — An equivalence relation
35. Let R be the relation {(1,1), (2,2), (3,3), (1,2), (2,1)} on A = {1,2,3}. R is: — Reflexive and symmetric, but not transitive
36. Consider the relation R = {(x, y) | x and y are integers and x - y is even} on the set of integers. R is: — An equivalence relation
37. An equivalence relation on a set A is a relation that is: — Reflexive, symmetric, and transitive
38. Let R be the relation 'has the same number of elements as' on the set of all finite sets. This relation is: — An equivalence relation
39. The set of all pairs (x, y) where x and y are integers and x ≥ y, forms a relation that is: — Reflexive, antisymmetric, and transitive
40. A relation R on a set A is a partial ordering if it is: — Reflexive, antisymmetric, and transitive
41. The relation 'is a sibling of' on the set of people is: — Symmetric and transitive, but not reflexive
42. The relation 'is a proper subset of' (⊂) on the set of all sets is: — Irreflexive, antisymmetric, and transitive
43. Consider the relation R = {(1,2), (2,1), (1,1)} on set A = {1,2}. R is: — Reflexive and symmetric, but not transitive
44. If A = {a, b, c} and R = {(a, a), (b, b), (c, c)}, R is: — Reflexive, symmetric, and transitive
45. Let R be the relation 'is married to' on the set of all people. R is: — Reflexive and symmetric, but not transitive
46. Which of the following is NOT a property of an equivalence relation? — Antisymmetry
47. Let R be the relation 'is less than' (<) on the set of real numbers. Which property does R NOT satisfy? — Reflexivity
48. Which property is required for a relation to be a partial ordering, but not necessarily for it to be an equivalence relation? — Antisymmetry
49. 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: — Two sets
50. 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? — No, because it is not transitive.
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