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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 — 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.
Q2 — 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 $\mathbb{N}$).
Q3 — 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: $A=B$.
Q4 — Venn Diagrams · hard · numerical
Among $600$ students, $300$ play cricket, $250$ hockey, $150$ football; $100$ cricket & hockey, $70$ hockey & football, $80$ cricket & football, and $50$ all three. How many play at least one?
A. $500$ ✓ Correct
B. $450$
C. $550$
D. $600$
Solution: Inclusion-exclusion: $(300+250+150) - (100+70+80) + 50 = 700 - 250 + 50 = 500$.
Q5 — Venn Diagrams · hard · numerical
For sets $A, B$ with $n(A - B) = 12$, $n(B - A) = 16$, and $n(A \cap B) = 8$, the value of $n(A \cup B)$ is:
A. $44$
B. $28$
C. $36$ ✓ Correct
D. $20$
Solution: $n(A\cup B) = n(A-B) + n(B-A) + n(A\cap B) = 12 + 16 + 8 = 36$.
Q6 — Subsets and Intervals · hard · numerical
How many subsets of $\{1, 2, 3, 4, 5, 6\}$ have their smallest element equal to $2$?
A. $16$ ✓ Correct
B. $4$
C. $8$
D. $32$
Solution: The subset must contain 2, must exclude 1, and may freely include any of $\{3,4,5,6\}$: $2^4 = 16$.
Q7 — Operations on Sets · hard · numerical
If $A$ and $B$ are disjoint with $n(A) = 7$ and $n(B) = 9$, then $n(A \cup B)$ equals:
A. $63$
B. $2$
C. $16$ ✓ Correct
D. $9$
Solution: For disjoint sets $n(A\cup B) = n(A) + n(B) = 7 + 9 = 16$ (since $n(A\cap B)=0$).
Q8 — Operations on Sets · hard · theory
The operation of set difference is:
A. not commutative in general ($A - B \ne B - A$) ✓ Correct
B. always commutative
C. always equal to the union
D. always empty
Solution: In general $A-B \ne B-A$; e.g. $\{1,2\}-\{2,3\}=\{1\}$ but $\{2,3\}-\{1,2\}=\{3\}$.
Q9 — Complement and De Morgan's Laws · hard · theory
For any set $A$, $A \cup (A' \cap B)$ simplifies to:
A. $A \cup B$ ✓ Correct
B. $A \cap B$
C. $A$
D. $B$
Solution: By distributivity: $A\cup(A'\cap B) = (A\cup A')\cap(A\cup B) = U\cap(A\cup B) = A\cup B$.
Q10 — Complement and De Morgan's Laws · hard · numerical
If $A$ and $B$ are subsets of $U$ with $n(U)=50$, $n(A')=30$, $n(B')=35$, and $n(A'\cap B')=15$, then $n(A\cap B)$ is:
A. $5$
B. $10$
C. $0$ ✓ Correct
D. $15$
Solution: $n(A)=20$, $n(B)=15$; $n(A\cup B)=n(U)-n(A'\cap B')=50-15=35$; $n(A\cap B)=20+15-35=0$.
Q11 — Sets and their Representations · hard · numerical
The number of elements in $\{x \in \mathbb{Z} : x^2 - 5|x| + 6 = 0\}$ is:
A. $2$
B. $4$ ✓ Correct
C. $3$
D. $1$
Solution: Let $t=|x|\ge0$: $t^2-5t+6=0 \Rightarrow t=2,3$, so $|x|=2$ or $3$, giving $x=\pm2,\pm3$: 4 values.
Q12 — The Empty Set · hard · theory
Which of the following is a non-empty set?
A. $\{x \in \mathbb{R} : |x| = -1\}$
B. $\{x \in \mathbb{N} : 2 < x < 3\}$
C. $\{x \in \mathbb{R} : x^2 + 1 = 0\}$
D. $\{x \in \mathbb{R} : x^2 - 2x + 1 = 0\}$ ✓ Correct
Solution: $x^2-2x+1=(x-1)^2=0$ has the real root $x=1$, so that set is $\{1\}$ — non-empty.
Q13 — The Empty Set · hard · theory
For how many real values of $k$ is $\{x \in \mathbb{R} : x^2 - 2x + k = 0\}$ empty?
A. exactly two
B. none
C. exactly one
D. infinitely many (all $k > 1$) ✓ Correct
Solution: No real roots when discriminant $4 - 4k < 0$, i.e. $k > 1$ — infinitely many such $k$.
Q14 — Finite and Infinite Sets · hard · theory
Which of the following sets is finite?
A. $\{x \in \mathbb{Q} : 0 < x < 1\}$
B. $\{x \in \mathbb{Z} : x > 5\}$
C. $\{x \in \mathbb{R} : \sin x = 0\}$
D. $\{x \in \mathbb{N} : x^2 - 7x + 12 = 0\}$ ✓ Correct
Solution: $x^2-7x+12=0 \Rightarrow x=3,4$ — a finite (two-element) set; the others are infinite.
Q15 — Finite and Infinite Sets · hard · theory
If $A$ is finite with $n$ elements and $B$ is infinite, then $A \cup B$ is:
A. empty
B. infinite ✓ Correct
C. finite
D. of exactly $n$ elements
Solution: A union containing an infinite set is infinite.
Q16 — Equal Sets · hard · theory
If $\{2a + b,\ a - b\} = \{5,\ 1\}$ with $a - b = 1$, then the value of $a$ is:
A. $2$ ✓ Correct
B. $4$
C. $3$
D. $1$
Solution: From the pairing $a-b=1$ and $2a+b=5$: adding, $3a=6 \Rightarrow a=2$ (then $b=1$).
Q17 — Subsets and Intervals · hard · numerical
The number of subsets of $\{1, 2, 3, 4, 5, 6, 7\}$ that contain both $1$ and $7$ is:
A. $32$ ✓ Correct
B. $31$
C. $16$
D. $64$
Solution: Fix 1 and 7 present; the other 5 elements are free: $2^5 = 32$.
Q18 — Power Set · hard · theory
If $A = \{\emptyset, \{\emptyset\}\}$, then the number of elements in $P(A)$ is:
A. $8$
B. $1$
C. $2$
D. $4$ ✓ Correct
Solution: $A$ has 2 elements ($\emptyset$ and $\{\emptyset\}$), so $n(P(A)) = 2^2 = 4$.
Q19 — Venn Diagrams · hard · numerical
In a group of $400$ people, $250$ speak Hindi, $200$ speak English. If everyone speaks at least one language, the number speaking both is:
A. $150$
B. $450$
C. $50$ ✓ Correct
D. $100$
Solution: Both $= 250 + 200 - 400 = 50$.
Q20 — Venn Diagrams · hard · numerical
In a class of $100$, $55$ passed maths, $67$ passed physics. What is the minimum possible number who passed both?
A. $0$
B. $12$
C. $33$
D. $22$ ✓ Correct
Solution: Minimum both occurs when the union is largest ($=100$): $n(\cap) \ge 55+67-100 = 22$.
Q21 — Operations on Sets · hard · theory
For any three sets, $A \cap (B - C)$ equals:
A. $(A - B) \cap C$
B. $(A \cap B) - C$ only when $C \subseteq A$
C. $A \cap B \cap C$
D. $(A \cap B) - (A \cap C)$ ✓ Correct
Solution: $A\cap(B-C) = A\cap B\cap C' = (A\cap B) - (A\cap C)$.
Q22 — Operations on Sets · hard · numerical
If $A = \{x \in \mathbb{R} : |x| \le 2\}$ and $B = \{x \in \mathbb{R} : 1 \le x \le 5\}$, then $A - B$ is:
A. $(1, 2]$
B. $[-2, 5]$
C. $[-2, 1]$
D. $[-2, 1)$ ✓ Correct
Solution: $A=[-2,2]$; removing $[1,5]$ leaves $[-2,1)$.
Q23 — Complement and De Morgan's Laws · hard · theory
The expression $(A \cap B') \cup (A \cap B)$ simplifies to:
A. $B$
B. $\emptyset$
C. $A$ ✓ Correct
D. $A \cup B$
Solution: $(A\cap B')\cup(A\cap B) = A\cap(B'\cup B) = A\cap U = A$.
Q24 — Sets and their Representations · medium · numerical
How many elements are in $\{x \in \mathbb{N} : x \text{ is a factor of } 36 \text{ and } x \text{ is even}\}$?
A. $4$
B. $3$
C. $5$
D. $6$ ✓ Correct
Solution: Even divisors of 36 are 2, 4, 6, 12, 18, 36 — six elements.
Q25 — Sets and their Representations · medium · numerical
The number of elements in $\{(x, y) : x, y \in \mathbb{N},\ x + y = 6\}$ is:
A. $5$ ✓ Correct
B. $4$
C. $7$
D. $6$
Solution: Ordered pairs of naturals summing to 6: $(1,5),(2,4),(3,3),(4,2),(5,1)$ = 5.
Q26 — The Empty Set · medium · theory
If $A = \{x \in \mathbb{Q} : x^2 = 5\}$, then $A$ is:
A. infinite
B. empty, since $\sqrt5$ is irrational ✓ Correct
C. a singleton
D. a two-element set
Solution: $\sqrt5$ is irrational, so no rational number squares to 5; $A$ is empty.
Q27 — Finite and Infinite Sets · medium · theory
A set is called finite if:
A. it is a subset of $\mathbb{N}$
B. it is empty or its elements can be counted, the counting ending after a definite number ✓ Correct
C. it has at least one element
D. it contains only integers
Solution: A finite set is empty or has a definite number $n$ of elements after which counting stops.
Q28 — Equal Sets · medium · theory
If $A = \{x : x \text{ is a multiple of } 2\}$ and $B = \{x : x \text{ is an even integer}\}$, then:
A. $A = B$ ✓ Correct
B. $A \subset B$ strictly
C. $B \subset A$ strictly
D. $A$ and $B$ are disjoint
Solution: "Multiple of 2" and "even integer" describe exactly the same numbers, so $A=B$.
Q29 — Power Set · medium · theory
For any set $A$, which relation is correct?
A. $A \in P(A)$ ✓ Correct
B. $A \subseteq P(A)$
C. $P(A) \in A$
D. $P(A) \subseteq A$
Solution: $A$ is a subset of itself, so $A$ is a member of its power set: $A \in P(A)$.
Q30 — Power Set · medium · numerical
If $n(P(A)) = 256$, then the number of elements in $A$ is:
A. $16$
B. $7$
C. $128$
D. $8$ ✓ Correct
Solution: $2^{n(A)} = 256 = 2^8 \Rightarrow n(A) = 8$.