TRB PG Assistant - Cardinal numbers - countable and uncountable cardinals, Cantor's diagonal process One Line Questions & Answers

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? — |Q| < |R|

2. What does ℵ₀ + ℵ₀ equal in terms of cardinal arithmetic? — ℵ₀

3. What does ℵ₀ * ℵ₀ equal in terms of cardinal arithmetic? — ℵ₀

4. Cantor's diagonal process is a constructive proof method. What does it construct? — A real number not present in a given list of real numbers.

5. 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? — Changing the nth digit of the nth number in the list.

6. The union of a finite number of countable sets is: — Countable

7. The union of a countable number of countable sets is: — Countable

8. What is the symbol for the cardinality of the set of natural numbers? — ℵ₀

9. The statement that there is no cardinality strictly between that of the natural numbers and the real numbers is known as: — The Continuum Hypothesis

10. What is the term used for a cardinal number that represents the size of a countably infinite set? — Aleph-null

11. Cantor's diagonal argument can be applied to show that the set of all functions from N to {0, 1} is: — Uncountable

12. The cardinality of the set of all transcendental numbers is: — Uncountable

13. If there exists a bijection (one-to-one correspondence) between set A and set B, then A and B have the same: — Cardinality

14. If a set is uncountable, it means it has a cardinality that is: — Greater than ℵ₀

15. What is the cardinality of the set of natural numbers (N)? — Countably infinite

16. What is the cardinality of the set of rational numbers (Q)? — Countably infinite

17. The set of all pairs of natural numbers (N x N) has which cardinality? — Countably infinite

18. What is the cardinality of the set of all finite strings over a finite alphabet (e.g., binary strings)? — Countably infinite

19. What is the cardinality of the set of all finite subsets of a countably infinite set? — Countably infinite

20. What is the cardinality of the set {1, 2, 3}? — 3

21. What is the cardinality of the empty set? — 0

22. What is the primary outcome of Cantor's diagonal argument concerning the set of real numbers? — It demonstrates that the set of real numbers is uncountable.

23. Cantor's diagonal process is a proof technique used to demonstrate what property of a set? — Its uncountability

24. What is the cardinality of the power set of a finite set with n elements? — 2^n

25. 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? — Only infinite sets

26. What is the fundamental concept introduced by Georg Cantor related to the sizes of sets? — Cardinal numbers

27. In the context of cardinal numbers, what does 'ℵ₁' represent? — The smallest uncountable cardinal number

28. 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? — The nth

29. What does it mean for a set to have 'uncountable' cardinality? — It is impossible to establish a one-to-one correspondence between the set and the natural numbers.

30. Which of the following sets is NOT countable? — The set of all real numbers (R)

31. The cardinality of the set of all integers (Z) is the same as the cardinality of which other set? — The set of all rational numbers (Q)

32. A set is called 'countable' if it can be put into a one-to-one correspondence with which set? — A subset of the natural numbers

33. Which of the following is a direct consequence of Cantor's diagonal argument applied to the set of real numbers? — The set of real numbers is uncountable.

34. The set of all points on a line segment (e.g., [0, 1]) has the same cardinality as: — The set of real numbers

35. Which of the following sets is countably infinite? — The set of all prime numbers

36. A set is 'countably infinite' if and only if it is equivalent to which set? — The set of natural numbers

37. 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: — There are infinitely many different sizes of infinity.

38. Which of the following is a property of countable sets? — They can be placed in one-to-one correspondence with a subset 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. — True

40. If a set A can be put into a one-to-one correspondence with a subset of the natural numbers, then A is: — Countable

41. The set of all finite sequences of natural numbers is: — Countably infinite

42. The set of all finite subsets of natural numbers is: — Countably infinite

43. The cardinality of the set of all algebraic numbers is: — Countably infinite

44. The set of all ordered pairs of integers (Z x Z) is: — Countably infinite

45. The cardinality of the set of all subsets of the natural numbers is: — c

46. 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)|? — c

47. What is the cardinality of the set of all possible infinite binary sequences? — c

48. What is the smallest uncountable cardinal number? — ℵ₁

49. The set of all functions from the set of natural numbers to itself (N^N) has which cardinality? — c

50. The cardinality of the set of real numbers (R) is denoted by which symbol? — c (continuum)

Cardinal numbers - countable and uncountable cardinals, Cantor's diagonal process - One Line Questions