Cardinal numbers - countable and uncountable cardinals, Cantor's diagonal process - Question Bank
1. What is the relationship between the cardinality of the set of rational numbers (Q) and the cardinality of the set of real numbers (R) according to Cantor?
2. Cantor's diagonal argument can be generalized to prove that for any set X, the cardinality of the power set P(X) is strictly greater than the cardinality of X. This implies:
3. If a set is 'Dedekind-infinite', it means it can be put into a one-to-one correspondence with a proper subset of itself. Which type of sets are Dedekind-infinite?
4. The set of all functions from the set of natural numbers to itself (N^N) has which cardinality?
5. What is the smallest uncountable cardinal number?
6. The union of a countable number of countable sets is:
7. The union of a finite number of countable sets is:
8. What is the cardinality of the set of all finite subsets of a countably infinite set?
9. Cantor's diagonal process is a constructive proof method. What does it construct?
10. The set of all ordered pairs of integers (Z x Z) is:
11. What is the cardinality of the set of all possible infinite binary sequences?
12. If a set is uncountable, it means it has a cardinality that is:
13. The set of all points on a line segment (e.g., [0, 1]) has the same cardinality as:
14. Which of the following is a property of countable sets?
15. What is the primary outcome of Cantor's diagonal argument concerning the set of real numbers?
16. A set is 'countably infinite' if and only if it is equivalent to which set?
17. Cantor's theorem states that for any set A, the cardinality of its power set P(A) is strictly greater than the cardinality of A. If |A| = ℵ₀, what is |P(A)|?
18. In the context of cardinal numbers, what does 'ℵ₁' represent?
19. The cardinality of the set of all transcendental numbers is:
20. The cardinality of the set of all algebraic numbers is:
21. What is the cardinality of the set of all finite strings over a finite alphabet (e.g., binary strings)?
22. Cantor's diagonal argument can be applied to show that the set of all functions from N to {0, 1} is:
23. The set of all pairs of natural numbers (N x N) has which cardinality?
24. What is the term used for a cardinal number that represents the size of a countably infinite set?
25. Consider an infinite list of decimal expansions of real numbers between 0 and 1. Cantor's diagonal process constructs a new real number that differs from the nth number in the list in which decimal place?
26. The set of all finite subsets of natural numbers is:
27. What is the cardinality of the empty set?
28. If there exists a bijection (one-to-one correspondence) between set A and set B, then A and B have the same:
29. Which of the following is a direct consequence of Cantor's diagonal argument applied to the set of real numbers?
30. The cardinality of the set of all subsets of the natural numbers is:
31. What does ℵ₀ * ℵ₀ equal in terms of cardinal arithmetic?
32. What does ℵ₀ + ℵ₀ equal in terms of cardinal arithmetic?
33. The set of all finite sequences of natural numbers is:
34. What is the cardinality of the set {1, 2, 3}?
35. If a set A can be put into a one-to-one correspondence with a subset of the natural numbers, then A is:
36. Which of the following sets is countably infinite?
37. The statement that there is no cardinality strictly between that of the natural numbers and the real numbers is known as:
38. What is the symbol for the cardinality of the set of natural numbers?
39. Cantor proved that the cardinality of the power set of any set S is strictly greater than the cardinality of S. This is known as Cantor's theorem.
40. What is the cardinality of the power set of a finite set with n elements?
41. In Cantor's diagonal argument for the real numbers, what is the crucial step that generates a new real number not present in an assumed list?
42. Cantor's diagonal process is a proof technique used to demonstrate what property of a set?
43. What does it mean for a set to have 'uncountable' cardinality?
44. The cardinality of the set of real numbers (R) is denoted by which symbol?
45. What is the cardinality of the set of rational numbers (Q)?
46. The cardinality of the set of all integers (Z) is the same as the cardinality of which other set?
47. What is the cardinality of the set of natural numbers (N)?
48. Which of the following sets is NOT countable?
49. A set is called 'countable' if it can be put into a one-to-one correspondence with which set?
50. What is the fundamental concept introduced by Georg Cantor related to the sizes of sets?