Learn › IISER · Maths ›
Sets › Subsets and Intervals
Subsets and Intervals — IISER Maths MCQs with Solutions
Free IISER Maths Subsets and Intervals MCQs with step-by-step solutions (22 questions). Part of Sets. Practise online on Prepizo — no login needed.
▶ Practise Subsets and Intervals online (free)
Questions with solutions
Q1 — 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; $2^4 = 16$.
Q2 — 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: $2^4 - 1 = 15$.
Q3 — 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.
Q4 — Subsets and Intervals · medium · theory
The interval $(2, 5]$ written in set-builder form is:
A. $\{x \in \mathbb{R} : 2 \le x < 5\}$
B. $\{x \in \mathbb{R} : 2 \le x \le 5\}$
C. $\{x \in \mathbb{R} : 2 < x < 5\}$
D. $\{x \in \mathbb{R} : 2 < x \le 5\}$ ✓ Correct
Solution: A round bracket excludes the endpoint, a square bracket includes it: $2 < x \le 5$.
Q5 — Subsets and Intervals · medium · theory
The set $\{x \in \mathbb{R} : -3 \le x < 7\}$ as an interval is:
A. $(-3, 7)$
B. $[-3, 7)$ ✓ Correct
C. $[-3, 7]$
D. $(-3, 7]$
Solution: Includes $-3$ (square bracket), excludes 7 (round bracket): $[-3,7)$.
Q6 — Subsets and Intervals · medium · theory
Which of the following is NOT a subset of $\{1, 2, 3, 4\}$?
A. $\{4\}$
B. $\emptyset$
C. $\{2, 5\}$ ✓ Correct
D. $\{1, 3\}$
Solution: $\{2,5\}$ contains 5, which is not in the set, so it is not a subset.
Q7 — Subsets and Intervals · medium · theory
Which symbol correctly fills the blank: $\{1, 2\}\ \_\_\ \{1, 2, 3\}$?
A. $\subset$ (proper subset) ✓ Correct
B. $=$
C. $\in$
D. $\supset$
Solution: $\{1,2\}$ is contained in $\{1,2,3\}$ and is not equal to it, so it is a proper subset.
Q8 — Subsets and Intervals · medium · theory
The number of elements in the interval $[3, 8]$ (as a subset of $\mathbb{R}$) is:
A. $8$
B. $5$
C. $6$
D. infinite ✓ Correct
Solution: A real interval contains infinitely many real numbers between its endpoints.
Q9 — Subsets and Intervals · medium · numerical
If $A = \{1, 2, 3\}$, how many subsets of $A$ contain the element $2$?
A. $3$
B. $2$
C. $8$
D. $4$ ✓ Correct
Solution: Fix 2 as present; the other two elements are free: $2^2 = 4$ subsets.
Q10 — Subsets and Intervals · medium · numerical
How many subsets of $\{1, 2, 3, 4, 5\}$ contain exactly $3$ elements?
A. $20$
B. $5$
C. $15$
D. $10$ ✓ Correct
Solution: Number of 3-element subsets $= \binom{5}{3} = 10$.
Q11 — Subsets and Intervals · hard · theory
If $A \subseteq B$ and $B \subseteq C$, then which is necessarily true?
A. $C \subseteq A$
B. $A \subseteq C$ ✓ Correct
C. $A$ and $C$ are disjoint
D. $A = C$
Solution: The subset relation is transitive: $A\subseteq B\subseteq C \Rightarrow A\subseteq C$.
Q12 — Subsets and Intervals · medium · theory
Which statement is correct?
A. The empty set has no subsets
B. A proper subset can equal the whole set
C. A set with 3 elements has 6 subsets
D. Every set is a subset of itself ✓ Correct
Solution: $A \subseteq A$ always holds; the empty set has one subset ($\emptyset$); a set of 3 elements has $2^3=8$ subsets.
Q13 — Subsets and Intervals · medium · numerical
The number of subsets of $\{1, 2, 3, 4, 5, 6\}$ that contain the element $1$ is:
A. $64$
B. $31$
C. $16$
D. $32$ ✓ Correct
Solution: Fix 1; the remaining 5 elements are free: $2^5 = 32$.
Q14 — Subsets and Intervals · medium · theory
The set $\{x \in \mathbb{R} : x \le 4\}$ as an interval is:
A. $[4, \infty)$
B. $(4, \infty)$
C. $(-\infty, 4)$
D. $(-\infty, 4]$ ✓ Correct
Solution: All reals up to and including 4: $(-\infty, 4]$.
Q15 — Subsets and Intervals · hard · numerical
The number of subsets of $\{1, 2, 3, 4, 5\}$ containing at least one even number is:
A. $16$
B. $8$
C. $24$ ✓ Correct
D. $28$
Solution: Total $2^5=32$; subsets with no even number (only odds $\{1,3,5\}$) $=2^3=8$; so $32-8=24$.
Q16 — Subsets and Intervals · medium · theory
Which interval represents $\{x \in \mathbb{R} : |x| < 3\}$?
A. $(0, 3)$
B. $[-3, 3]$
C. $(-\infty, 3)$
D. $(-3, 3)$ ✓ Correct
Solution: $|x|<3 \Leftrightarrow -3 < x < 3$, i.e. the open interval $(-3,3)$.
Q17 — Subsets and Intervals · medium · numerical
If a set has exactly $63$ proper subsets, the number of elements it has is:
A. $63$
B. $7$
C. $5$
D. $6$ ✓ Correct
Solution: $2^n - 1 = 63 \Rightarrow 2^n = 64 \Rightarrow n = 6$.
Q18 — Subsets and Intervals · medium · theory
Which of the following relations is FALSE?
A. $\{1\} \subseteq \{1, 2, 3\}$
B. $\emptyset \subseteq \{1, 2, 3\}$
C. $1 \in \{1, 2, 3\}$
D. $\{1\} \in \{1, 2, 3\}$ ✓ Correct
Solution: $\{1\}$ is a subset, not an element, of $\{1,2,3\}$, so $\{1\}\in\{1,2,3\}$ is false.
Q19 — Subsets and Intervals · medium · theory
The intervals $[a, b]$ and $(a, b)$ differ in that:
A. they are equal
B. $[a,b]$ excludes both endpoints
C. $[a, b]$ includes both endpoints while $(a, b)$ excludes both ✓ Correct
D. $(a,b)$ includes both endpoints
Solution: Square brackets include endpoints; round brackets exclude them.
Q20 — 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 $\{3,4,5,6\}$: $2^4 = 16$.
Q21 — 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: $2^5 = 32$.
Q22 — Subsets and Intervals · medium · numerical
How many subsets of $\{1, 2, 3, 4, 5\}$ contain neither $1$ nor $2$?
A. $16$
B. $6$
C. $4$
D. $8$ ✓ Correct
Solution: Such subsets are subsets of $\{3,4,5\}$: $2^3 = 8$.