The Empty Set — IISER Maths MCQs with Solutions
Free IISER Maths The Empty Set MCQs with step-by-step solutions (14 questions). Part of Sets. Practise online on Prepizo — no login needed.
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Questions with solutions
Q1 — The Empty Set · medium · theory
Which of the following sets is empty?
A. $\{x \in \mathbb{N} : x < 3\}$
B. $\{x \in \mathbb{R} : x^2 + 4 = 0\}$ ✓ Correct
C. $\{x \in \mathbb{R} : x^2 = 4\}$
D. $\{x \in \mathbb{Z} : x^2 = 9\}$
Solution: $x^2+4=0$ has no real solution (needs $x^2=-4$), so the set is empty.
Q2 — The Empty Set · medium · theory
Which of the following sets is NOT empty?
A. $\{x \in \mathbb{R} : x^2 + 1 = 0\}$
B. $\{x \in \mathbb{Q} : x^2 = 3\}$
C. $\{x \in \mathbb{R} : x^2 = 2\}$ ✓ Correct
D. $\{x \in \mathbb{Z} : 1 < x < 2\}$
Solution: $x^2=2$ has real solutions $\pm\sqrt2$, so that set is non-empty. The others are empty (no integer strictly between 1 and 2; $x^2=-1$ has no real root; $\sqrt3$ is irrational).
Q3 — 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 $n(\emptyset)=0$.
Q4 — The Empty Set · medium · theory
Which statement is correct?
A. $\emptyset = \{0\}$
B. $\{0\}$ has no elements
C. $\{0\}$ is not the empty set because it contains the element $0$ ✓ Correct
D. $\{0\}$ is the empty set
Solution: $\{0\}$ is a set containing one element (the number 0), so it is non-empty; $\emptyset$ contains nothing.
Q5 — 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.
Q6 — The Empty Set · medium · theory
The set $\{x \in \mathbb{N} : x < 1\}$ is:
A. a singleton set
B. an infinite set
C. equal to $\{0\}$
D. the empty set ✓ Correct
Solution: No natural number is less than 1 (naturals start at 1), so the set is empty. (Even with 0 excluded, it is empty.)
Q7 — The Empty Set · medium · theory
Which set is empty?
A. $\{x : x \text{ is an odd prime}\}$
B. $\{x : x \text{ is an even prime greater than } 2\}$ ✓ Correct
C. $\{x : x \text{ is an even prime}\}$
D. $\{x : x \text{ is a prime less than } 5\}$
Solution: 2 is the only even prime, so there is no even prime greater than 2 — the set is empty.
Q8 — The Empty Set · medium · theory
The set $\{x \in \mathbb{R} : x^2 - x + 1 = 0\}$ is:
A. a singleton
B. empty ✓ Correct
C. infinite
D. a two-element set
Solution: Discriminant $= 1 - 4 = -3 < 0$, so there are no real roots; the set is empty.
Q9 — The Empty Set · medium · theory
The set of points common to two distinct parallel lines in a plane is:
A. the empty set ✓ Correct
B. a pair of points
C. a single point
D. infinite
Solution: Distinct parallel lines never intersect, so their common-point set is empty.
Q10 — The Empty Set · medium · theory
Which of these represents the empty set?
A. $\{x : x \ne x\}$ ✓ Correct
B. $\{\emptyset\}$
C. $\{x : x = x\}$
D. $\{x : x^2 \ge 0\}$
Solution: No object satisfies $x \ne x$, so $\{x : x \ne x\}$ is empty.
Q11 — 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.
Q12 — 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.
Q13 — 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.
Q14 — 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 $k$.