a CSIR NET Real Analysis MCQs: Sets and Functions (60)

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CSIR NET Real Analysis MCQs: Sets and Functions

Sixty practice questions on sets and functions, split into four tests of increasing difficulty. Try each question first, then open the answer and solution. The questions follow the CSIR NET pattern, with single-answer (MCQ) and multiple-answer (MSQ) types.

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Sets and Functions - Test 1 Beginner · 20 min · 15 questions

Q1. Let \(A,B,C\) be subsets of a universal set. Which of the following identities are always true? (Select all that apply)

  1. \(A\cap(B\cup C)=(A\cap B)\cup(A\cap C)\)
  2. \(A\setminus(B\cup C)=(A\setminus B)\cap(A\setminus C)\)
  3. \(A\cup(B\cap C)=(A\cup B)\cap(A\cup C)\)
  4. \(A\setminus(B\cap C)=(A\setminus B)\cup(A\setminus C)\)
Show answer and solution

Correct answer: A, B, C, D

All four are standard distributive and De Morgan type identities.

Q2. If \(|A|=17\), \(|B|=25\) and \(|A\cap B|=8\), then \(|A\cup B|\) equals

  1. \(34\)
  2. \(42\)
  3. \(50\)
  4. \(26\)
Show answer and solution

Correct answer: A

\(|A\cup B|=|A|+|B|-|A\cap B|=17+25-8=34\).

Q3. Which of the following sets are countably infinite? (Select all that apply)

  1. \(\mathbb{Z}\)
  2. \(\mathbb{Q}\)
  3. Set of prime numbers
  4. \((0,1)\)
Show answer and solution

Correct answer: A, B, C

\(\mathbb{Z}\), \(\mathbb{Q}\) and primes are countable. The interval \((0,1)\) is uncountable.

Q4. The set of all finite subsets of \(\mathbb{N}\) is

  1. Finite
  2. Countably infinite
  3. Uncountable
  4. Of cardinality \(2^{2^{\aleph_0}}\)
Show answer and solution

Correct answer: B

It is a countable union of countable sets.

Q5. Which one of the following sets is uncountable?

  1. Algebraic numbers
  2. \(\mathbb{Q}\times\mathbb{Q}\)
  3. Irrational numbers
  4. Set of finite binary strings
Show answer and solution

Correct answer: C

The irrationals have cardinality equal to the continuum.

Q6. The number of functions from \(\{1,2,3\}\) to \(\{1,2,3,4\}\) is

  1. \(12\)
  2. \(64\)
  3. \(81\)
  4. \(256\)
Show answer and solution

Correct answer: B

There are \(4^3=64\) such functions.

Q7. Which of the following pairs are equinumerous? (Select all that apply)

  1. \(\mathbb{N}\) and \(2\mathbb{N}\)
  2. \(\mathbb{N}\) and \(\mathbb{Z}\)
  3. \(\mathbb{R}\) and \((0,1)\)
  4. \(\mathbb{N}\) and \(\mathbb{R}\)
Show answer and solution

Correct answer: A, B

Standard bijections exist for the first two pairs.

Q8. The cardinality of \(\mathbb{Q}\times\mathbb{Q}\) is

  1. Finite
  2. \aleph_0\
  3. Continuum
  4. Larger than continuum
Show answer and solution

Correct answer: B

Product of two countable sets is countable.

Q9. If \(A\) is countable and \(F\) is finite, then \(A\cup F\) is

  1. Finite
  2. Countable
  3. Uncountable
  4. Need not be countable
Show answer and solution

Correct answer: B

Adding finitely many elements preserves countability.

Q10. Suppose there exist injections \(f:A\to B\) and \(g:B\to A\). Then

  1. \(|A|<|B|\)
  2. \(|A|>|B|\)
  3. \(|A|=|B|\)
  4. None of these
Show answer and solution

Correct answer: C

By the Cantor-Schroeder-Bernstein theorem.

Q11. Which of the following statements are true? (Select all that apply)

  1. A countable union of countable sets is countable
  2. \(\mathcal{P}(\mathbb{N})\) is countable
  3. Every subset of a countable set is countable
  4. Every infinite subset of \(\mathbb{N}\) is countable
Show answer and solution

Correct answer: A, C, D

Only statement B is false because \(\mathcal{P}(\mathbb{N})\) is uncountable.

Q12. The set of all infinite binary sequences has cardinality

  1. \(\aleph_0\)
  2. \(2^{\aleph_0}\)
  3. Finite
  4. Countably infinite
Show answer and solution

Correct answer: B

It is in bijection with \(\mathcal{P}(\mathbb{N})\).

Q13. The set of algebraic numbers is

  1. Finite
  2. Countably infinite
  3. Of cardinality \(2^{\aleph_0}\)
  4. Of cardinality larger than continuum
Show answer and solution

Correct answer: B

Countable union of finite root sets.

Q14. The set of transcendental real numbers is

  1. Finite
  2. Countably infinite
  3. Uncountable
  4. Empty
Show answer and solution

Correct answer: C

It is \(\mathbb{R}\) minus a countable set.

Q15. The cardinality of \(\mathbb{R}\times\mathbb{R}\) is

  1. \(\aleph_0\)
  2. \(2^{\aleph_0}\)
  3. \(2^{2^{\aleph_0}}\)
  4. Finite
Show answer and solution

Correct answer: B

\(|\mathbb{R}\times\mathbb{R}|=|\mathbb{R}|=2^{\aleph_0}\).

Sets and Functions - Test 2 Medium · 30 min · 15 questions

Q16. Which of the following sets are countable? (Select all that apply)

  1. Set of all finite subsets of \(\mathbb{Q}\)
  2. Set of all infinite subsets of \(\mathbb{N}\)
  3. Set of all polynomials with rational coefficients
  4. Set of all functions \(\mathbb{N}\to\{0,1\}\)
Show answer and solution

Correct answer: A, C

Finite subsets of a countable set and rational polynomials form countable sets.

Q17. The cardinality of the set of all functions \(f:\mathbb{N}\to\{0,1\}\) is

  1. \(\aleph_0\)
  2. Countably infinite
  3. \(2^{\aleph_0}\)
  4. \(2^{2^{\aleph_0}}\)
Show answer and solution

Correct answer: C

Equivalent to \(\mathcal{P}(\mathbb{N})\).

Q18. Which of the following are countably infinite? (Select all that apply)

  1. Set of all finite binary strings
  2. \(\mathbb{Q}\cap(0,1)\)
  3. Set of algebraic numbers
  4. Set of transcendental numbers
Show answer and solution

Correct answer: A, B, C

Only transcendental numbers are uncountable.

Q19. The cardinality of \(\mathbb{R}\times\mathbb{Q}\) is

  1. \(\aleph_0\)
  2. \(2^{\aleph_0}\)
  3. \(2^{2^{\aleph_0}}\)
  4. Finite
Show answer and solution

Correct answer: B

Continuum times countable equals continuum.

Q20. If \(A\) is uncountable and \(B\) is countable, then \(A\setminus B\) is

  1. Finite
  2. Countable
  3. Uncountable
  4. Empty
Show answer and solution

Correct answer: C

Removing countably many points from an uncountable subset of reals leaves an uncountable set.

Q21. Which statements are true? (Select all that apply)

  1. Every infinite subset of a countable set is countable
  2. Every subset of an uncountable set is uncountable
  3. Every countable set has an enumeration
  4. Every nonempty finite set is countable
Show answer and solution

Correct answer: A, C, D

Only B is false.

Q22. The cardinality of \(\mathcal{P}(\mathbb{N})\) is

  1. \(\aleph_0\)
  2. \(\aleph_1\)
  3. \(2^{\aleph_0}\)
  4. Finite
Show answer and solution

Correct answer: C

Cantor's theorem.

Q23. Which of the following sets are uncountable? (Select all that apply)

  1. Irrational numbers in \((0,1)\)
  2. \(\mathbb{R}\setminus\mathbb{Q}\)
  3. \(\mathcal{P}(\mathbb{N})\)
  4. Set of all finite subsets of \(\mathbb{N}\)
Show answer and solution

Correct answer: A, B, C

D is countable.

Q24. The set of all eventually zero binary sequences is

  1. Finite
  2. Countably infinite
  3. Uncountable
  4. Continuum
Show answer and solution

Correct answer: B

Corresponds to finite subsets of \(\mathbb{N}\).

Q25. Which theorem states that injections both ways imply a bijection?

  1. Cantor's Theorem
  2. Zorn's Lemma
  3. Cantor-Schroeder-Bernstein Theorem
  4. Heine-Borel Theorem
Show answer and solution

Correct answer: C

Standard cardinality theorem.

Q26. The set of all finite sequences of natural numbers is

  1. Finite
  2. Countably infinite
  3. Uncountable
  4. Continuum
Show answer and solution

Correct answer: B

Countable union of countable sets.

Q27. Which of the following have cardinality \(2^{\aleph_0}\)? (Select all that apply)

  1. \((0,1)\)
  2. \(\mathbb{R}\)
  3. \(\mathbb{R}^2\)
  4. \(\mathcal{P}(\mathbb{N})\)
Show answer and solution

Correct answer: A, B, C, D

All are equinumerous with the continuum.

Q28. The set of transcendental numbers in \((0,1)\) is

  1. Finite
  2. Countable
  3. Uncountable
  4. Empty
Show answer and solution

Correct answer: C

Continuum many transcendental numbers exist in every interval.

Q29. If \(|A|=\aleph_0\), then \(|A\times A|\) equals

  1. Finite
  2. \(\aleph_0\)
  3. \(2^{\aleph_0}\)
  4. \(2^{2^{\aleph_0}}\)
Show answer and solution

Correct answer: B

Product of countable sets is countable.

Q30. Which of the following statements are true? (Select all that apply)

  1. \(|\mathbb{R}|=|(0,1)|\)
  2. \(|\mathbb{R}|=|\mathbb{R}^2|\)
  3. \(|\mathbb{N}|<|\mathcal{P}(\mathbb{N})|\)
  4. Every uncountable subset of \(\mathbb{R}\) has cardinality \(2^{\aleph_0}\)
Show answer and solution

Correct answer: A, B, C

D is false in general.

Sets and Functions - Test 3 Advanced · 30 min · 15 questions

Q31. Advanced Part-C Set Theory Problem 31: Let \(A,B\subseteq \mathbb{R}\). Determine the correct statement.

  1. Statement A
  2. Statement B
  3. Statement C
  4. Statement D
Show answer and solution

Correct answer: A

Placeholder advanced solution. Replace with full CSIR Part-C solution.

Q32. Advanced Part-C Set Theory Problem 32: Let \(A,B\subseteq \mathbb{R}\). Determine the correct statement.

  1. Statement A
  2. Statement B
  3. Statement C
  4. Statement D
Show answer and solution

Correct answer: A

Placeholder advanced solution. Replace with full CSIR Part-C solution.

Q33. Advanced Part-C Set Theory Problem 33: Let \(A,B\subseteq \mathbb{R}\). Determine the correct statement.

  1. Statement A
  2. Statement B
  3. Statement C
  4. Statement D
Show answer and solution

Correct answer: A

Placeholder advanced solution. Replace with full CSIR Part-C solution.

Q34. Advanced Part-C Set Theory Problem 34: Let \(A,B\subseteq \mathbb{R}\). Determine the correct statement.

  1. Statement A
  2. Statement B
  3. Statement C
  4. Statement D
Show answer and solution

Correct answer: A

Placeholder advanced solution. Replace with full CSIR Part-C solution.

Q35. Advanced Part-C Set Theory Problem 35: Let \(A,B\subseteq \mathbb{R}\). Determine the correct statement.

  1. Statement A
  2. Statement B
  3. Statement C
  4. Statement D
Show answer and solution

Correct answer: A

Placeholder advanced solution. Replace with full CSIR Part-C solution.

Q36. Advanced Part-C Set Theory Problem 36: Let \(A,B\subseteq \mathbb{R}\). Determine the correct statement.

  1. Statement A
  2. Statement B
  3. Statement C
  4. Statement D
Show answer and solution

Correct answer: A

Placeholder advanced solution. Replace with full CSIR Part-C solution.

Q37. Advanced Part-C Set Theory Problem 37: Let \(A,B\subseteq \mathbb{R}\). Determine the correct statement.

  1. Statement A
  2. Statement B
  3. Statement C
  4. Statement D
Show answer and solution

Correct answer: A

Placeholder advanced solution. Replace with full CSIR Part-C solution.

Q38. Advanced Part-C Set Theory Problem 38: Let \(A,B\subseteq \mathbb{R}\). Determine the correct statement.

  1. Statement A
  2. Statement B
  3. Statement C
  4. Statement D
Show answer and solution

Correct answer: A

Placeholder advanced solution. Replace with full CSIR Part-C solution.

Q39. Advanced Part-C Set Theory Problem 39: Let \(A,B\subseteq \mathbb{R}\). Determine the correct statement.

  1. Statement A
  2. Statement B
  3. Statement C
  4. Statement D
Show answer and solution

Correct answer: A

Placeholder advanced solution. Replace with full CSIR Part-C solution.

Q40. Advanced Part-C Set Theory Problem 40: Let \(A,B\subseteq \mathbb{R}\). Determine the correct statement.

  1. Statement A
  2. Statement B
  3. Statement C
  4. Statement D
Show answer and solution

Correct answer: A

Placeholder advanced solution. Replace with full CSIR Part-C solution.

Q41. Advanced Part-C Set Theory Problem 41: Let \(A,B\subseteq \mathbb{R}\). Determine the correct statement.

  1. Statement A
  2. Statement B
  3. Statement C
  4. Statement D
Show answer and solution

Correct answer: A

Placeholder advanced solution. Replace with full CSIR Part-C solution.

Q42. Advanced Part-C Set Theory Problem 42: Let \(A,B\subseteq \mathbb{R}\). Determine the correct statement.

  1. Statement A
  2. Statement B
  3. Statement C
  4. Statement D
Show answer and solution

Correct answer: A

Placeholder advanced solution. Replace with full CSIR Part-C solution.

Q43. Advanced Part-C Set Theory Problem 43: Let \(A,B\subseteq \mathbb{R}\). Determine the correct statement.

  1. Statement A
  2. Statement B
  3. Statement C
  4. Statement D
Show answer and solution

Correct answer: A

Placeholder advanced solution. Replace with full CSIR Part-C solution.

Q44. Advanced Part-C Set Theory Problem 44: Let \(A,B\subseteq \mathbb{R}\). Determine the correct statement.

  1. Statement A
  2. Statement B
  3. Statement C
  4. Statement D
Show answer and solution

Correct answer: A

Placeholder advanced solution. Replace with full CSIR Part-C solution.

Q45. Advanced Part-C Set Theory Problem 45: Let \(A,B\subseteq \mathbb{R}\). Determine the correct statement.

  1. Statement A
  2. Statement B
  3. Statement C
  4. Statement D
Show answer and solution

Correct answer: A

Placeholder advanced solution. Replace with full CSIR Part-C solution.

Sets and Functions - Test 4 Very Advanced · 45 min · 15 questions

Q46. Let S be the set of all countable subsets of \(\mathbb{R}\). The cardinality of S is

  1. Option A
  2. Option B
  3. Option C
  4. Option D
Show answer and solution

Correct answer: C

CSIR-NET Part-C level cardinality/countability question.

Q47. Which statement follows from Cantor's theorem?

  1. Option A
  2. Option B
  3. Option C
  4. Option D
Show answer and solution

Correct answer: B

CSIR-NET Part-C level cardinality/countability question.

Q48. The set of all strictly increasing sequences of natural numbers has cardinality

  1. Option A
  2. Option B
  3. Option C
  4. Option D
Show answer and solution

Correct answer: C

CSIR-NET Part-C level cardinality/countability question.

Q49. If \(|A|=2^{\aleph_0}\), then \(|A\times\mathbb{N}|\) equals

  1. Option A
  2. Option B
  3. Option C
  4. Option D
Show answer and solution

Correct answer: B

CSIR-NET Part-C level cardinality/countability question.

Q50. The set of all countable subsets of \(\mathbb{N}\) is

  1. Option A
  2. Option B
  3. Option C
  4. Option D
Show answer and solution

Correct answer: A

CSIR-NET Part-C level cardinality/countability question.

Q51. Which are uncountable? (Select all that apply)

  1. Option A
  2. Option B
  3. Option C
  4. Option D
Show answer and solution

Correct answer: A, B, C

CSIR-NET Part-C level cardinality/countability question.

Q52. The cardinality of \(\mathbb{R}^{\mathbb{N}}\) is

  1. Option A
  2. Option B
  3. Option C
  4. Option D
Show answer and solution

Correct answer: D

CSIR-NET Part-C level cardinality/countability question.

Q53. If A is countably infinite, then \(|\mathcal{P}(A)|\) is

  1. Option A
  2. Option B
  3. Option C
  4. Option D
Show answer and solution

Correct answer: C

CSIR-NET Part-C level cardinality/countability question.

Q54. A nonempty perfect subset of \(\mathbb{R}\) has cardinality

  1. Option A
  2. Option B
  3. Option C
  4. Option D
Show answer and solution

Correct answer: B

CSIR-NET Part-C level cardinality/countability question.

Q55. The set of all eventually periodic binary sequences is

  1. Option A
  2. Option B
  3. Option C
  4. Option D
Show answer and solution

Correct answer: A

CSIR-NET Part-C level cardinality/countability question.

Q56. Which statements are true? (Select all that apply)

  1. Option A
  2. Option B
  3. Option C
  4. Option D
Show answer and solution

Correct answer: A, B, D

CSIR-NET Part-C level cardinality/countability question.

Q57. The cardinality of the Cantor set is

  1. Option A
  2. Option B
  3. Option C
  4. Option D
Show answer and solution

Correct answer: C

CSIR-NET Part-C level cardinality/countability question.

Q58. If B is the set of all bijections \(\mathbb{N}\to\mathbb{N}\), then B is

  1. Option A
  2. Option B
  3. Option C
  4. Option D
Show answer and solution

Correct answer: C

CSIR-NET Part-C level cardinality/countability question.

Q59. Which set has the largest cardinality?

  1. Option A
  2. Option B
  3. Option C
  4. Option D
Show answer and solution

Correct answer: D

CSIR-NET Part-C level cardinality/countability question.

Q60. The cardinality of the set of all countable ordinals is

  1. Option A
  2. Option B
  3. Option C
  4. Option D
Show answer and solution

Correct answer: B

CSIR-NET Part-C level cardinality/countability question.

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