Set Theory Questions with Solutions

31 questions19 easy10 medium2 hard

A set is a well-defined collection of objects. Set theory covers types of sets (finite, infinite, empty, singleton, equal and equivalent sets), subsets and power sets, and operations on sets such as union, intersection, difference and complement. It also covers De Morgan's laws and the cardinality formula n(A union B) = n(A) + n(B) - n(A intersection B). In entrance exams, the most common questions are word problems about people who like one, two or three activities, solved with Venn diagrams and the cardinality formula. Questions also ask for the number of subsets of a set (2^n for a set with n elements) and test set notation and properties. Practise the set theory questions below. Each comes with a step-by-step solution and a Venn diagram where it helps.

All Questions Easy Medium Hard

Q1.
The set $(A Δ B) \cap C$ is not equal to:
A. $(A \cap C) Δ (B \cap C)$ B. $((A \cap B^{c}) \cap C) \cup ((B \cap A^{c}) \cap C)$ C. $(A \cap B)^{c} \cap C$ D. $(A \cap B \cap C)^{c} \cap ((A \cup B) \cap C)$
mediumAsked in NIMCET 2026View solution ›
Q2.
If there are 5 sets represented via a Venn diagram, the number of regions that arise in the Venn diagram equals:
A. 22B. 20C. 28D. 32
easyAsked in NIMCET 2026View solution ›
Q3.
Let $R \subset N \times N$ . Which of the following statements is necessarily true?
A. If for each $a \in N$ the set $R_{a} := b \in N : (a, b) \in R$ has cardinality at most 1, then $R$ represents a function $f : N \to N$ B. If for each $b \in N$ the set $R^{b} := a \in N : (a, b) \in R$ has cardinality at most 1, then $R$ represents a function $f : N \to N$ C. If for some $a \in N$ the set $R_{a}$ is infinite, then $R_{a}$ and $R$ have the same cardinalityD. If for all $a \in N$ the set $R_{a}$ is infinite, then the cardinality of $R$ is larger than that of $R_{a}$
hardAsked in NIMCET 2026View solution ›
Q4.
The number of triples of sets $(A, B, C)$ with $A, B, C \subseteq 1, \dots, n$ such that $(A \cap B) \subseteq C \subseteq (A \cup B)$ is:
A. $5^{n}$ B. $6^{n}$ C. $7^{n}$ D. $8^{n}$
hardAsked in NIMCET 2026View solution ›
Q5.
If $X = 4^{n} - 3 n - 1 ∣ n \in N$ and $Y = 9 (n - 1) ∣ n \in N$ , then
A. $X \subset Y$ B. $Y \subset X$ C. $X = Y$ D. None of these
mediumAsked in AMU MCA 2016View solution ›
Q6.
In a Venn diagram, what does the overlapping region of two sets $A$ and $B$ represent?
A. Union of A and BB. Intersection of A and BC. Complement of AD. Difference between A and B
easyAsked in MAH MCA CET 2025View solution ›
Q7.
Let A and B be sets such that $A \cap X = B \cap X = ϕ$ and $A \cup X = B \cup X$ for some set X. Then
A. $A = B$ B. $A \cup B = X$ C. $B = X$ D. $A = X$
easyAsked in NIMCET 2023View solution ›
Q8.
Find the number of elements in the union of 4 sets A, B, C and D having 150, 180, 210 and 240 elements respectively, given that each pair of sets has 15 elements in common, each triple of sets has 3 elements in common and $A \cap B \cap C \cap D = \emptyset$ .
A. 616B. 512C. 111D. 702
easyAsked in NIMCET 2013View solution ›
Q9.
Let $X$ be the universal set for sets $A$ and $B$ . If $n (A) = 200$ , $n (B) = 300$ and $n (A \cap B) = 100$ , then $n (A' \cap B') = 300$ provided $n (X)$ is equal to
A. $600$ B. $700$ C. $800$ D. $900$
easyAsked in NIMCET 2011View solution ›
Q10.
In a college of $300$ students, every student reads $5$ newspapers and every newspaper is read by $60$ students. The number of newspapers is
A. at least $30$ B. at most $20$ C. exactly $25$ D. exactly $28$
easyAsked in NIMCET 2011View solution ›
Q11.
How many proper subsets of $1, 2, 3, 4, 5, 6, 7$ contain the numbers $1$ and $7$ ?
A. 7B. 31C. 32D. 62
easyAsked in NIMCET 2010View solution ›
Q12.
A set contains $(2 n + 1)$ elements. If the number of subsets that contain at most $n$ elements is $4096$ , then the value of $n$ is:
A. 28B. 21C. 15D. 6
mediumAsked in NIMCET 2009View solution ›
Q13.
If $P = 4^{n} - 3 n - 1 : n \in N$ and $Q = 9 n - 9 : n \in N$ , then $P \cup Q$ equals:
A. $N$ B. $P$ C. $Q$ D. None of these
mediumAsked in NIMCET 2009View solution ›
Q14.
$A_{1}, A_{2}, A_{3}, A_{4}$ are subsets of a set $U$ containing 75 elements. Each $A_{i}$ has 28 elements, any two of them have 12 elements in common, any three have 5 elements in common and all four have 1 element in common. The number of elements belonging to none of the four subsets is:
A. 15B. 17C. 16D. 18
mediumAsked in NIMCET 2009View solution ›
Q15.
Suppose $P_{1}, P_{2}, \dots, P_{30}$ are thirty sets each having $5$ elements and $Q_{1}, Q_{2}, \dots, Q_{n}$ are $n$ sets with $3$ elements each. Let $⋃_{i = 1}^{30} P_{i} = ⋃_{j = 1}^{n} Q_{j} = S$ and each element of $S$ belongs to exactly $10$ of the $P_{i}$ 's and exactly $9$ of the $Q_{j}$ 's. Then $n$ equals
A. $15$ B. $3$ C. $45$ D. None of these
mediumAsked in NIMCET 2008View solution ›
Q16.
If A is a subset of B and B is a subset of C, then cardinality of $A \cup B \cup C$ is equal to
A. Cardinality of CB. Cardinality of BC. Cardinality of AD. None of the above
easyAsked in NIMCET 2020View solution ›
Q17.
In a class of 50 students, it was found that 30 students read "Hitavad", 35 students read "Hindustan" and 10 read neither. How many students read both "Hitavad" and "Hindustan" newspapers?
A. 25B. 20C. 15D. 30
easyAsked in NIMCET 2020View solution ›
Q18.
If $A = x, y, z$ , then the number of subsets in powerset of A is
A. 6B. 8C. 7D. 9
easyAsked in NIMCET 2020View solution ›
Q19.
If $A = 1, 2, 3, 4$ and $B = 3, 4, 5$ , then the number of elements in $(A \cup B) \times (A \cap B) \times (A Δ B)$ is
A. 18B. 20C. 24D. 30
easyAsked in NIMCET 2021View solution ›
Q20.
Suppose $A_{1}, A_{2}, \dots, A_{30}$ are 30 sets, each with five elements, and $B_{1}, B_{2}, \dots, B_{n}$ are $n$ sets, each with three elements, such that $⋃_{i = 1}^{30} A_{i} = ⋃_{j = 1}^{n} B_{j} = S$ , and each element of S belongs to exactly ten of the $A_{i}$ 's and exactly nine of the $B_{j}$ 's. Then $n =$
A. 15B. 45C. 75D. 90
mediumAsked in NIMCET 2021 +1View solution ›

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