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Sets — IISER Maths MCQs with Solutions
Free IISER Maths Sets MCQs with step-by-step solutions covering Sets and their Representations, The Empty Set, Finite and Infinite Sets, Equal Sets, Subsets and Intervals, Power Set. Practise online on Prepizo — no login needed.
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Sample questions with solutions
Q1 — Sets and their Representations · easy · numerical
The set $\{x : x \text{ is a prime number and } x < 20\}$ in roster form is:
A. $\{3,5,7,11,13,17,19\}$
B. $\{2,3,5,7,11,13,17,19\}$ ✓ Correct
C. $\{1,2,3,5,7,11,13,17,19\}$
D. $\{2,3,5,7,9,11,13,17,19\}$
Solution: Primes below 20: 2, 3, 5, 7, 11, 13, 17, 19 (1 is not prime, 9 is composite).
Q2 — Sets and their Representations · easy · numerical
The roster form of $\left\{ \dfrac{n}{n+1} : n \in \mathbb{N},\ n \le 4 \right\}$ is:
A. $\left\{\tfrac12, \tfrac23, \tfrac34, \tfrac45\right\}$ ✓ Correct
B. $\left\{\tfrac11, \tfrac12, \tfrac13, \tfrac14\right\}$
C. $\left\{1, 2, 3, 4\right\}$
D. $\left\{\tfrac12, \tfrac23, \tfrac34, \tfrac45, \tfrac56\right\}$
Solution: For $n=1,2,3,4$: $\tfrac12,\tfrac23,\tfrac34,\tfrac45$.
Q3 — Sets and their Representations · easy · numerical
The roster form of $\{x \in \mathbb{Z} : -3 < x \le 2\}$ is:
A. $\{-2, -1, 0, 1\}$
B. $\{-2, -1, 0, 1, 2\}$ ✓ Correct
C. $\{-3, -2, -1, 0, 1, 2\}$
D. $\{-3, -2, -1, 0, 1\}$
Solution: Integers strictly greater than $-3$ and at most $2$: $-2,-1,0,1,2$.
Q4 — The Empty Set · easy · theory
The number of elements in the empty set $\emptyset$ is:
A. undefined
B. $0$ ✓ Correct
C. $1$
D. infinite
Solution: The empty (null) set has no elements, so $n(\emptyset)=0$.
Q5 — Finite and Infinite Sets · easy · theory
Which of the following is an infinite set?
A. $\{x \in \mathbb{N} : x < 100\}$
B. $\{x \in \mathbb{Z} : x^2 < 50\}$
C. $\{x \in \mathbb{R} : 0 < x < 1\}$ ✓ Correct
D. $\{x : x \text{ is a letter of the English alphabet}\}$
Solution: There are infinitely many real numbers between 0 and 1; the other three are finite.
Q6 — Equal Sets · easy · theory
Which pair of sets is equal?
A. $\{1, 2, 3\}$ and $\{1, 2, 4\}$
B. $\{1, 2\}$ and $\{1, 2, 3\}$
C. $\{1, 2, 3\}$ and $\{3, 2, 1\}$ ✓ Correct
D. $\{1\}$ and $\{1, 1, 1\}$ only by counting
Solution: Two sets are equal iff they have exactly the same elements; order and repetition do not matter, so $\{1,2,3\}=\{3,2,1\}$.
Q7 — Power Set · easy · numerical
If $A = \{1, 2\}$, then $P(A)$ (the power set of $A$) is:
A. $\{\emptyset, \{1\}, \{2\}, \{1, 2\}\}$ ✓ Correct
B. $\{\{1\}, \{2\}, \{1, 2\}\}$
C. $\{1, 2, \{1, 2\}\}$
D. $\{\{1\}, \{2\}\}$
Solution: The power set is the set of all subsets, including $\emptyset$ and $A$ itself: four subsets.
Q8 — Power Set · easy · numerical
If a set $A$ has $5$ elements, then the number of elements in $P(A)$ is:
A. $25$
B. $10$
C. $32$ ✓ Correct
D. $16$
Solution: $n(P(A)) = 2^{n(A)} = 2^5 = 32$.
Q9 — Universal Set · easy · theory
The universal set for a particular discussion is:
A. a set that contains all objects under consideration in that context ✓ Correct
B. the empty set
C. always an infinite set
D. always the set of natural numbers
Solution: The universal set $U$ contains all the elements relevant to the particular context being studied.
Q10 — Venn Diagrams · easy · theory
In a Venn diagram, the universal set is usually represented by:
A. a circle
B. a rectangle ✓ Correct
C. a triangle
D. a point
Solution: By convention the universal set is drawn as a rectangle, with its subsets as circles inside.
Q11 — Subsets and Intervals · easy · numerical
The number of subsets of a set with $4$ elements is:
A. $4$
B. $8$
C. $16$ ✓ Correct
D. $15$
Solution: A set with $n$ elements has $2^n$ subsets; $2^4 = 16$.
Q12 — Subsets and Intervals · easy · numerical
The number of proper subsets of $\{a, b, c, d\}$ is:
A. $8$
B. $14$
C. $16$
D. $15$ ✓ Correct
Solution: Proper subsets exclude the set itself: $2^4 - 1 = 15$.
Q13 — Subsets and Intervals · easy · theory
For any set $A$, which is always true?
A. $\emptyset \in A$
B. $A \subset A$ (strictly)
C. $\emptyset \subseteq A$ ✓ Correct
D. $A \in A$
Solution: The empty set is a subset of every set; and every set is a subset (not proper) of itself.
Q14 — Operations on Sets · easy · numerical
If $A = \{1, 2, 3, 4\}$ and $B = \{3, 4, 5, 6\}$, then $A \cup B$ is:
A. $\{1, 2, 3, 4, 5, 6\}$ ✓ Correct
B. $\{1, 2, 5, 6\}$
C. $\{3, 4\}$
D. $\{1, 2, 3, 4\}$
Solution: The union collects all elements from either set: $\{1,2,3,4,5,6\}$.
Q15 — Operations on Sets · easy · numerical
If $A = \{1, 2, 3, 4\}$ and $B = \{3, 4, 5, 6\}$, then $A \cap B$ is:
A. $\{1, 2, 3, 4, 5, 6\}$
B. $\{1, 2, 5, 6\}$
C. $\emptyset$
D. $\{3, 4\}$ ✓ Correct
Solution: The intersection is the common elements: $\{3,4\}$.
Q16 — Operations on Sets · easy · numerical
If $A = \{1, 2, 3, 4\}$ and $B = \{3, 4, 5, 6\}$, then $A - B$ is:
A. $\{3, 4\}$
B. $\{1, 2\}$ ✓ Correct
C. $\{5, 6\}$
D. $\{1, 2, 5, 6\}$
Solution: $A - B$ keeps elements of $A$ not in $B$: $\{1,2\}$.
Q17 — Operations on Sets · easy · theory
If $A \subseteq B$, then $A \cup B$ equals:
A. $A$
B. $B$ ✓ Correct
C. $A \cap B$
D. $\emptyset$
Solution: When $A$ is contained in $B$, the union is the larger set $B$.
Q18 — Operations on Sets · easy · theory
If $A \subseteq B$, then $A \cap B$ equals:
A. $B$
B. $A \cup B$
C. $A$ ✓ Correct
D. $\emptyset$
Solution: When $A\subseteq B$, the common part is all of $A$: $A\cap B = A$.
Q19 — Complement and De Morgan's Laws · easy · numerical
If $U = \{1,2,3,4,5,6,7,8\}$ and $A = \{1,3,5,7\}$, then $A'$ is:
A. $\{2,4,6,8\}$ ✓ Correct
B. $U$
C. $\{1,2,3,4\}$
D. $\{1,3,5,7\}$
Solution: $A' = U - A = \{2,4,6,8\}$.
Q20 — Complement and De Morgan's Laws · easy · theory
For any set $A$ with universal set $U$, the value of $(A')'$ is:
A. $U$
B. $A$ ✓ Correct
C. $\emptyset$
D. $A'$
Solution: Complementing twice gives back the original set: $(A')' = A$.
Q21 — Complement and De Morgan's Laws · easy · theory
The complement of the universal set, $U'$, is:
A. undefined
B. $U$
C. $\emptyset$ ✓ Correct
D. the power set of $U$
Solution: $U' = U - U = \emptyset$.
Q22 — Complement and De Morgan's Laws · easy · theory
The complement of the empty set, $\emptyset'$, is:
A. $U$ ✓ Correct
B. $A$
C. a singleton
D. $\emptyset$
Solution: $\emptyset' = U - \emptyset = U$.
Q23 — Sets and their Representations · hard · numerical
The number of ordered pairs in the set $\{(x, y) : x, y \in \mathbb{Z},\ x^2 + y^2 = 25\}$ is:
A. $4$
B. $12$ ✓ Correct
C. $8$
D. $16$
Solution: Solutions: $(\pm5,0),(0,\pm5),(\pm3,\pm4),(\pm4,\pm3)$ = $2+2+4+4 = 12$.
Q24 — The Empty Set · hard · theory
Which of the following is true about $\{\emptyset\}$?
A. It is the empty set
B. It equals $\emptyset$
C. It is a non-empty set containing one element, namely $\emptyset$ ✓ Correct
D. It has zero elements
Solution: $\{\emptyset\}$ has exactly one member (the empty set), so $n(\{\emptyset\})=1$; it is not itself empty.
Q25 — The Empty Set · hard · theory
The number of subsets of the empty set $\emptyset$ is:
A. $2$
B. $1$ ✓ Correct
C. $0$
D. infinite
Solution: The only subset of $\emptyset$ is $\emptyset$ itself, so it has $2^0 = 1$ subset.
Q26 — Finite and Infinite Sets · hard · theory
If $A$ is an infinite set, then its power set $P(A)$ is:
A. a singleton
B. infinite ✓ Correct
C. empty
D. finite
Solution: The power set of an infinite set is infinite (in fact of strictly greater cardinality).
Q27 — Finite and Infinite Sets · hard · theory
The set of rational numbers $\mathbb{Q}$ is:
A. finite
B. empty
C. uncountable
D. infinite (countably infinite) ✓ Correct
Solution: $\mathbb{Q}$ is infinite; moreover it is countably infinite (can be put in one-to-one correspondence with $\mathbb{N}$).
Q28 — Equal Sets · hard · theory
If $\{a, b\} = \{2, 3\}$ and $a < b$, then $2a + b$ equals:
A. $6$
B. $8$
C. $5$
D. $7$ ✓ Correct
Solution: From $a<b$: $a=2, b=3$, so $2a+b = 4+3 = 7$.
Q29 — Equal Sets · hard · theory
Sets $A$ and $B$ satisfy $A \subseteq B$ and $B \subseteq A$. Then:
A. $A \subset B$ strictly
B. nothing can be concluded
C. $A$ and $B$ are disjoint
D. $A = B$ ✓ Correct
Solution: Mutual inclusion $A\subseteq B$ and $B\subseteq A$ is precisely the definition of set equality: $A=B$.
Q30 — Power Set · hard · theory
If $A = \{1, \{2\}\}$, which statement is TRUE?
A. $2 \in A$
B. $\{1\} \in A$
C. $\{2\} \in A$ and $\{\{2\}\} \in P(A)$ ✓ Correct
D. $\{2\} \subseteq A$
Solution: $A$ has elements $1$ and $\{2\}$; so $\{2\}\in A$, and $\{\{2\}\}$ is a subset of $A$, hence in $P(A)$.