Null Set, Universal Set, Subset and Proper Subset MCQs: 12 Solved Questions

Students often know the definitions but still lose marks by treating ∅, {∅} and {0} as the same object. The other common mistake is swapping the symbols for element, subset and proper subset. Write one line of reasoning before each answer; that forces the symbol check before intuition. For wider subject-wise revision, use GATE CS Exam Preparation.

Null set, universal set and subset rules to use before the MCQs

Calibrate with U = {1, 2, 3, 4}, A = {1, 2} and B = {1, 2, 3}. We have ∅ ⊆ A, A ⊂ B and B ⊂ U. Relative to U, Aᶜ = U - A = {3, 4} and ∅ᶜ = U.

Also, P(A) = {∅, {1}, {2}, {1, 2}}. Since |A| = 2, A has 2² = 4 subsets and 2² - 1 = 3 proper subsets.

Use these two checks throughout: count the braces before deciding membership, and ask whether equality is allowed before finding a maximum from a subset statement.

Null set and universal set MCQs 1 to 4

Question 1: Which of the following is a null set?

Which of the following is a null set?

• A. {0}

• B. { }

• C. {1, 2, 3}

• D. {x | x > 5}

Answer: B. { }

Count elements. {0} has one, {1, 2, 3} has three, and option D is non-empty over the usual numerical domains. Only { } has zero. Option D depends on an unstated domain; over the usual number domains it is non-empty. Option B is unambiguously empty.

Question 2: What is a null set or empty set?

What is a null set or empty set?

• A. A set with infinite elements

• B. A set with one element

• C. A set with zero elements

• D. A set with all the possible elements

Answer: C. A set with zero elements

A null set E satisfies |E| = 0. A singleton has cardinality 1. A universal set contains everything in the current domain.

Question 3: A={x |x≠x} represents

A={x |x≠x} represents

• A. {x}

• B. {1}

• C. {}

• D. {0}

Answer: C. {}

Every object equals itself, so x ≠ x selects nothing. Thus |A| = 0, not a singleton containing x.

Question 4: What is the universal set in Set Theory?

What is the universal set in Set Theory?

• A. The set of all sets

• B. The empty set

• C. The set of all subsets

• D. The set of all elements

Answer: D. The set of all elements

A universal set contains everything in the stated domain. It is neither a set of all sets nor a power set of subsets.

Subset and proper subset MCQs 5 to 8

Question 5: What is a proper subset?

What is a proper subset?

• A. A subset that includes all elements of the set

• B. A subset that includes some but not all elements of the set

• C. A subset that includes no elements of the set

• D. A subset that includes exactly one element of the set

Answer: B. A subset that includes some but not all elements of the set

The exact rule is A ⊆ B and A ≠ B. Since ∅ ⊂ {1} although ∅ contains nothing. More exactly, the condition is A ⊆ B and A ≠ B.

Question 6: Which of the following sets has no proper subset?

Which of the following sets has no proper subset?

• A. {1}

• B. { }

• C. {1, 2}

• D. Universal set

Answer: B. { }

The empty set’s only subset is itself, so it has no proper subset. The singleton {1}, however, has the proper subset ∅.

Question 7: Which of the following is the correct representation of a subset?

Which of the following is the correct representation of a subset?

• A. A ⊂ B

• B. A ⊃ B

• C. A ∩ B

• D. A ∪ B

Answer: A. A ⊂ B

This item uses A ⊂ B for containment. Many books use ⊆ generally and reserve ⊂ for proper subsets, so check the paper’s convention. The other symbols mean superset, intersection and union.

Question 8: If A is a subset of B and B is a subset of A, what can be said about A and B?

If A is a subset of B and B is a subset of A, what can be said about A and B?

• A. A is a proper subset of B

• B. B is a proper subset of A

• C. A = B

• D. A and B are disjoint

Answer: C. A = B

Each inclusion rules out elements outside the other set. Both contain exactly the same elements, so antisymmetry gives A = B.

Complement, inclusion and nested-set MCQs 9 to 12

Question 9: What is the complement of a null set in the universal set U?

What is the complement of a null set in the universal set U?

• A. { }

• B. U

• C. Null set

• D. Universal set

Answer: B. U

Directly, ∅ᶜ = U - ∅ = U. Options B and D are mathematically identical because both name U. Choose B only because it is the symbolic form; the mathematical result is ∅ᶜ = U.

Question 10: If A ⊂ B, then which of the following is not true?

If A ⊂ B, then which of the following is not true?

• A. A U B = B

• B. A ⋂ B = A

• C. Bᶜ ⊂ Aᶜ

• D. B – A = ɸ

Answer: D. B – A = ɸ

For U = {1, 2, 3}, A = {1} and B = {1, 2}, A ∪ B = B, A ∩ B = A, and Bᶜ = {3} ⊂ Aᶜ = {2, 3}. But B - A = {2}, so D is false. Here, A U B means union.

Question 11: Maximum cardinality sum in a nested chain

Consider a sequence of n sets defined as follows: B1 ⊆ B2 ⊆ B3 ⊆ … ⊆ Bn. The cardinality of the set Bn is n. What is the maximum value of |B1| + |B2| + … + |Bn|?

• A. n

• B. n²

• C. n(n+1)/2

• D. None of these

Answer: B. n²

Non-strict inclusion lets every Bᵢ equal Bₙ. Each size is at most n, so the sum is at most n + n + ... + n = n². For n = 4, let every set be {1, 2, 3, 4}. The sum reaches 4 + 4 + 4 + 4 = 16, not 10.

Question 12: Membership and subset statements

This question appeared in UGC NET 2025, Computer Science, Paper 2 (January).

Which of the following statements are true about the sets.

A. 0 ∈ ∅

B. ∅ ∈ {0}

C. ∅ ∈ {∅}

D. {∅} ∈ {∅}

E. {∅} ⊂ {∅, {∅}}

Choose the correct answer from the options given below:

• A. A, B, C, D and E

• B. A, B, C and E only

• C. A, C only

• D. C and E only

Answer: D. C and E only

A is false because ∅ has no elements. B is false because {0} contains only 0. C is true because {∅} contains ∅. D is false because it contains ∅, not {∅}. E is true because the right set contains the left set’s sole element plus another. Only C and E survive.

The five traps these set-theory MCQs expose

Quick diagnostic: braces around ∅ give cardinality 1, ⊆ permits equal adjacent sets, and complements need U first. Question 5 needs A ⊆ B and A ≠ B; Question 9 has two options with the same meaning.

A 10-minute retry method and related practice

For each of Questions 3, 6, 10, 11 and 12, spend 20 seconds identifying the symbol, 30 seconds testing U = {1, 2, 3}, and 10 seconds checking cardinality or equality. Use the next 60 seconds to answer again without the options and state the decisive rule. Five two-minute retries fill ten minutes. The sequence moves through emptiness, zero proper subsets, a counterexample, non-strict inclusion, and membership versus subset notation.

Use Set Theory and Relations Explained for GATE when a definition is shaky. Then solve Set Theory and Relations MCQs: 12 Solved (GATE). Move to the GATE Test Series only after you can explain every answer aloud.

Null set and subset MCQs: the short version and next step

∅ has no elements, every set contains ∅ as a subset, a proper subset is contained but unequal, mutual inclusion gives equality, and U depends on context. Without notes, reproduce U = {1, 2, 3, 4}, A = {1, 2}, B = {1, 2, 3} and the n = 4 maximum. If either fails, retry that question before moving to practice. GATE Guidance by Sanchit Sir places Discrete Mathematics beside the rest of the CS syllabus. Next, solve the wider set-theory MCQs, then take a mixed Discrete Mathematics set.