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Sets — JEE Main Mathematics MCQs with Solutions
Free JEE Main Mathematics 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. $\{2,3,5,7,9,11,13,17,19\}$
B. $\{2,3,5,7,11,13,17,19\}$ ✓ Correct
C. $\{1,2,3,5,7,11,13,17,19\}$
D. $\{3,5,7,11,13,17,19\}$
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. $\left\{\tfrac11, \tfrac12, \tfrac13, \tfrac14\right\}$
B. $\left\{\tfrac12, \tfrac23, \tfrac34, \tfrac45\right\}$ ✓ Correct
C. $\left\{\tfrac12, \tfrac23, \tfrac34, \tfrac45, \tfrac56\right\}$
D. $\left\{1, 2, 3, 4\right\}$
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. $\{-3, -2, -1, 0, 1\}$
B. $\{-2, -1, 0, 1, 2\}$ ✓ Correct
C. $\{-3, -2, -1, 0, 1, 2\}$
D. $\{-2, -1, 0, 1\}$
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. infinite
D. $1$
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. $\{x \in \mathbb{R} : 0 < x < 1\}$ ✓ Correct
B. $\{x \in \mathbb{Z} : x^2 < 50\}$
C. $\{x \in \mathbb{N} : x < 100\}$
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. $\{1\}$ and $\{1, 1, 1\}$ only by counting
C. $\{1, 2\}$ and $\{1, 2, 3\}$
D. $\{1, 2, 3\}$ and $\{3, 2, 1\}$ ✓ Correct
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. $\{1, 2, \{1, 2\}\}$
B. $\{\emptyset, \{1\}, \{2\}, \{1, 2\}\}$ ✓ Correct
C. $\{\{1\}, \{2\}\}$
D. $\{\{1\}, \{2\}, \{1, 2\}\}$
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. $16$
B. $25$
C. $32$ ✓ Correct
D. $10$
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. always an infinite set
C. the empty 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 point
B. a rectangle ✓ Correct
C. a circle
D. a triangle
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. $15$
B. $16$ ✓ Correct
C. $8$
D. $4$
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. $15$ ✓ Correct
C. $16$
D. $14$
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. $A \in A$
B. $\emptyset \subseteq A$ ✓ Correct
C. $A \subset A$ (strictly)
D. $\emptyset \in A$
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. $\{1, 2, 5, 6\}$
B. $\{3, 4\}$
C. $\{1, 2, 3, 4, 5, 6\}$ ✓ Correct
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. $\{3, 4\}$ ✓ Correct
B. $\emptyset$
C. $\{1, 2, 3, 4, 5, 6\}$
D. $\{1, 2, 5, 6\}$
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. $\{5, 6\}$
B. $\{3, 4\}$
C. $\{1, 2\}$ ✓ Correct
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. $A \cap B$
B. $\emptyset$
C. $A$
D. $B$ ✓ Correct
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. $\emptyset$
C. $A \cup B$
D. $A$ ✓ Correct
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. $\{1,2,3,4\}$
B. $U$
C. $\{1,3,5,7\}$
D. $\{2,4,6,8\}$ ✓ Correct
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. $\emptyset$
C. $A$ ✓ Correct
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. $\emptyset$ ✓ Correct
B. undefined
C. the power set of $U$
D. $U$
Solution: $U' = U - U = \emptyset$.
Q22 — Complement and De Morgan's Laws · easy · theory
The complement of the empty set, $\emptyset'$, is:
A. a singleton
B. $\emptyset$
C. $A$
D. $U$ ✓ Correct
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. $8$
C. $16$
D. $12$ ✓ Correct
Solution: Solutions: $(\pm5,0),(0,\pm5),(\pm3,\pm4),(\pm4,\pm3)$ = $2+2+4+4 = 12$.
Q24 — Sets and their Representations · hard · numerical
The number of elements in the set $\{x \in \mathbb{Z} : |x - 3| + |x + 2| = 5\}$ is:
A. $5$
B. $6$ ✓ Correct
C. infinite
D. $2$
Solution: $|x-3|+|x+2|=5$ holds for all $x$ in $[-2,3]$ (the distance between $-2$ and $3$). Integers there: $-2,-1,0,1,2,3$ = 6.
Q25 — Sets and their Representations · hard · numerical
If $A = \{x \in \mathbb{R} : x = \tfrac{2t}{1+t^2},\ t \in \mathbb{R}\}$, then $A$ equals:
A. $(-1, 1)$
B. $\mathbb{R}$
C. $[-1, 1]$ ✓ Correct
D. $[0, 1]$
Solution: With $x=\sin(2\theta)$ substitution ($t=\tan\theta$), $x$ ranges over $[-1,1]$; endpoints are attained at $t=\pm1$.
Q26 — Sets and their Representations · hard · numerical
The number of elements in $\{x \in \mathbb{Z} : \tfrac{x-2}{x+3} \in \mathbb{Z}\}$ is:
A. $2$
B. $5$
C. infinite
D. $4$ ✓ Correct
Solution: $\tfrac{x-2}{x+3}=1-\tfrac{5}{x+3}$; integer iff $(x+3)\mid5$, so $x+3\in\{\pm1,\pm5\}$, i.e. $x=-2,-4,2,-8$: 4 values.
Q27 — The Empty Set · hard · theory
Which of the following is true about $\{\emptyset\}$?
A. It has zero elements
B. It is a non-empty set containing one element, namely $\emptyset$ ✓ Correct
C. It is the empty set
D. It equals $\emptyset$
Solution: $\{\emptyset\}$ has exactly one member (the empty set), so $n(\{\emptyset\})=1$; it is not itself empty.
Q28 — The Empty Set · hard · numerical
For how many integer values of $k$ is the set $\{x \in \mathbb{R} : x^2 + kx + 1 = 0\}$ empty?
A. $2$
B. $1$
C. $3$ ✓ Correct
D. infinitely many
Solution: Empty (no real roots) when discriminant $k^2 - 4 < 0$, i.e. $-2 < k < 2$; integer $k = -1, 0, 1$: three values.
Q29 — The Empty Set · hard · theory
The set $\{x \in \mathbb{R} : \sin x = 2\}$ is:
A. empty ✓ Correct
B. a singleton
C. infinite
D. equal to $\{2\}$
Solution: $\sin x \in [-1,1]$ for all real $x$, so $\sin x = 2$ has no solution — the set is empty.
Q30 — Finite and Infinite Sets · hard · theory
If $A$ is an infinite set, then its power set $P(A)$ is:
A. finite
B. a singleton
C. infinite ✓ Correct
D. empty
Solution: The power set of an infinite set is infinite (in fact of strictly greater cardinality).