TRB PG Assistant - Cardinal numbers - countable and uncountable cardinals, Cantor's diagonal process Online Test (MCQ Quiz)

1. What is the fundamental concept introduced by Georg Cantor related to the sizes of sets?

A) Set equivalence B) Cardinal numbers C) Ordinal numbers D) Power sets

2. A set is called 'countable' if it can be put into a one-to-one correspondence with which set?

A) The set of all real numbers B) The set of all irrational numbers C) A subset of the natural numbers D) The empty set

3. Which of the following sets is NOT countable?

A) The set of all integers (Z) B) The set of all rational numbers (Q) C) The set of all natural numbers (N) D) The set of all real numbers (R)

4. What is the cardinality of the set of natural numbers (N)?

A) Finite B) Countably infinite C) Uncountably infinite D) Zero

5. The cardinality of the set of all integers (Z) is the same as the cardinality of which other set?

A) The set of all rational numbers (Q) B) The set of all real numbers (R) C) The set of all prime numbers D) The set of all positive integers

6. What is the cardinality of the set of rational numbers (Q)?

A) Finite B) Countably infinite C) Uncountably infinite D) Aleph-null plus one

7. The cardinality of the set of real numbers (R) is denoted by which symbol?

A) ℵ₀ (aleph-null) B) c (continuum) C) ω (omega) D) ℵ₁ (aleph-one)

8. What does it mean for a set to have 'uncountable' cardinality?

A) The set is empty. B) The set has a finite number of elements. C) It is impossible to establish a one-to-one correspondence between the set and the natural numbers. D) The set is a subset of the integers.

9. Cantor's diagonal process is a proof technique used to demonstrate what property of a set?

A) Its finiteness B) Its countability C) Its uncountability D) Its emptiness

10. 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?

A) Adding 1 to each digit of the diagonal elements. B) Changing the nth digit of the nth number in the list. C) Reversing the order of the digits. D) Multiplying each digit by a prime number.

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

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

12. 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.

A) True B) False C) True, but only for finite sets D) False, for infinite sets

13. What is the symbol for the cardinality of the set of natural numbers?

A) c B) ℵ₁ C) ℵ₀ D) ℵ₂

14. The statement that there is no cardinality strictly between that of the natural numbers and the real numbers is known as:

A) Cantor's Theorem B) The Continuum Hypothesis C) The Diagonal Argument D) The Axiom of Choice

15. Which of the following sets is countably infinite?

A) The set of points on a line segment B) The set of all transcendental numbers C) The set of all prime numbers D) The set of all real numbers between 0 and 1

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

A) Uncountable B) Countable C) Finite D) Empty

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

A) Infinite B) Countable C) 3 D) Aleph-null

18. The set of all finite sequences of natural numbers is:

A) Uncountable B) Countably infinite C) Finite D) Incomparable with N

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

A) 2ℵ₀ B) ℵ₀ C) ℵ₁ D) Finite

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

A) 2ℵ₀ B) ℵ₀ C) ℵ₁ D) Infinite

21. The cardinality of the set of all subsets of the natural numbers is:

A) ℵ₀ B) c C) ℵ₁ D) Finite

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

A) The set of integers is uncountable. B) The set of rational numbers is uncountable. C) The set of real numbers is uncountable. D) The set of prime numbers is infinite.

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

A) Elements B) Order C) Cardinality D) Structure

24. What is the cardinality of the empty set?

A) Infinite B) Countable C) 0 D) Aleph-null

25. The set of all finite subsets of natural numbers is:

A) Uncountable B) Countably infinite C) Finite D) Larger than R

26. 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?

A) The first B) The last C) The nth D) All of them

27. What is the term used for a cardinal number that represents the size of a countably infinite set?

A) Continuum B) Aleph-null C) Omega D) Finite

28. The set of all pairs of natural numbers (N x N) has which cardinality?

A) Finite B) Uncountable C) Countably infinite D) Same as R

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

A) Countably infinite B) Finite C) Uncountable D) Empty

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

A) Finite B) Uncountable C) Countably infinite D) Equal to ℵ₁

31. The cardinality of the set of all algebraic numbers is:

A) Uncountable B) Finite C) Countably infinite D) Equal to c

32. The cardinality of the set of all transcendental numbers is:

A) Countably infinite B) Finite C) Uncountable D) Zero

33. In the context of cardinal numbers, what does 'ℵ₁' represent?

A) The cardinality of the natural numbers B) The smallest uncountable cardinal number C) The cardinality of the real numbers D) The cardinality of the continuum

34. 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)|?

A) ℵ₀ B) ℵ₁ C) c D) Finite

35. A set is 'countably infinite' if and only if it is equivalent to which set?

A) The set of real numbers B) The set of integers C) The set of natural numbers D) The set of rational numbers

36. What is the primary outcome of Cantor's diagonal argument concerning the set of real numbers?

A) It proves that every real number can be represented as a fraction. B) It establishes that the set of real numbers is countable. C) It demonstrates that the set of real numbers is uncountable. D) It shows that there are different sizes of infinity.

37. Which of the following is a property of countable sets?

A) They are always finite. B) They can be placed in one-to-one correspondence with a subset of natural numbers. C) They contain an infinite number of elements. D) They are equivalent to the set of real numbers.

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

A) The set of natural numbers B) The set of integers C) The set of real numbers D) The set of rational numbers

39. If a set is uncountable, it means it has a cardinality that is:

A) Equal to 0 B) Finite and non-zero C) Greater than ℵ₀ D) Equal to ℵ₀

40. What is the cardinality of the set of all possible infinite binary sequences?

A) ℵ₀ B) c C) ℵ₁ D) Finite

41. The set of all ordered pairs of integers (Z x Z) is:

A) Uncountable B) Finite C) Countably infinite D) Larger than R

42. Cantor's diagonal process is a constructive proof method. What does it construct?

A) A list of all real numbers. B) A real number not present in a given list of real numbers. C) A bijection between N and R. D) A finite set.

43. What is the cardinality of the set of all finite subsets of a countably infinite set?

A) Finite B) Uncountable C) Countably infinite D) Equal to the original set's cardinality

44. The union of a finite number of countable sets is:

A) Always uncountable B) Always finite C) Countable D) Sometimes uncountable

45. The union of a countable number of countable sets is:

A) Always uncountable B) Always finite C) Countable D) Sometimes uncountable

46. What is the smallest uncountable cardinal number?

A) ℵ₀ B) c C) ℵ₁ D) ω

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

A) ℵ₀ B) c C) ℵ₁ D) Uncountable, but less than the power set of R

48. 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?

A) Only finite sets B) Only uncountable sets C) Only infinite sets D) Only empty sets

49. 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:

A) There is a largest cardinal number. B) There are infinitely many different sizes of infinity. C) All infinite sets have the same cardinality. D) The continuum hypothesis is false.

50. 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?

A) |Q| > |R| B) |Q| < |R| C) |Q| = |R| D) |Q| is finite, |R| is infinite

Cardinal numbers - countable and uncountable cardinals, Cantor's diagonal process - Online Test

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