Power Set — IISER Maths MCQs with Solutions
Free IISER Maths Power Set MCQs with step-by-step solutions (13 questions). Part of Sets. Practise online on Prepizo — no login needed.
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Questions with solutions
Q1 — 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.
Q2 — 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: $n(P(A)) = 2^{n(A)} = 2^5 = 32$.
Q3 — Power Set · medium · numerical
If $n(P(A)) = 128$, then $n(A)$ equals:
A. $8$
B. $64$
C. $6$
D. $7$ ✓ Correct
Solution: $2^{n(A)} = 128 = 2^7 \Rightarrow n(A) = 7$.
Q4 — Power Set · medium · theory
Which of the following is always TRUE for any set $A$?
A. $\emptyset \in P(A)$ and $A \in P(A)$ ✓ Correct
B. $A \subseteq P(A)$
C. $P(A) = A$
D. $\emptyset \subseteq P(A)$ only
Solution: Both $\emptyset$ and $A$ are subsets of $A$, hence members of $P(A)$.
Q5 — 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 $P(A)$.
Q6 — Power Set · medium · numerical
The number of elements in $P(P(\emptyset))$ is:
A. $1$
B. $4$
C. $2$ ✓ Correct
D. $0$
Solution: $P(\emptyset) = \{\emptyset\}$ has 1 element, so $P(P(\emptyset))$ has $2^1 = 2$ elements.
Q7 — Power Set · hard · numerical
The number of elements in $P(P(P(\emptyset)))$ is:
A. $16$
B. $2$
C. $4$ ✓ Correct
D. $8$
Solution: $n(\emptyset)=0 \to n(P(\emptyset))=1 \to n(P(P(\emptyset)))=2 \to n(P(P(P(\emptyset)))) = 2^2 = 4$.
Q8 — Power Set · medium · numerical
If $A = \{a, b, c\}$, the number of proper subsets of $A$ is:
A. $6$
B. $8$
C. $3$
D. $7$ ✓ Correct
Solution: Proper subsets exclude $A$ itself: $2^3 - 1 = 7$.
Q9 — Power Set · medium · theory
Which is FALSE for $A = \{1, 2, 3\}$?
A. $\{1\} \in P(A)$
B. $1 \in P(A)$ ✓ Correct
C. $\emptyset \in P(A)$
D. $\{1, 2\} \in P(A)$
Solution: Elements of $P(A)$ are subsets of $A$. $1$ is a number, not a subset, so $1 \in P(A)$ is false (though $\{1\}\in P(A)$ is true).
Q10 — 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: $A \in P(A)$.
Q11 — 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: $2^{n(A)} = 256 = 2^8 \Rightarrow n(A) = 8$.
Q12 — 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 $n(P(A)) = 2^2 = 4$.
Q13 — Power Set · medium · theory
For sets $A \subseteq B$, which relation between their power sets holds?
A. $P(A) = P(B)$
B. $P(A) \subseteq P(B)$ ✓ Correct
C. $P(B) \subseteq P(A)$
D. $P(A) \cap P(B) = \emptyset$
Solution: Every subset of $A$ is also a subset of $B$, so $P(A) \subseteq P(B)$.