Learn › JEE Main · Mathematics ›
Sets › Power Set
Power Set — JEE Main Mathematics MCQs with Solutions
Free JEE Main Mathematics Power Set MCQs with step-by-step solutions (13 questions). Part of Sets. Practise online on Prepizo — no login needed.
▶ Practise Power Set online (free)
Questions with solutions
Q1 — 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.
Q2 — 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: $n(P(A)) = 2^{n(A)} = 2^5 = 32$.
Q3 — Power Set · medium · numerical
If $n(P(A)) = 128$, then $n(A)$ equals:
A. $8$
B. $6$
C. $7$ ✓ Correct
D. $64$
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. $\emptyset \subseteq P(A)$ only
D. $P(A) = A$
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. $\{1\} \in A$
B. $2 \in A$
C. $\{2\} \subseteq A$
D. $\{2\} \in A$ and $\{\{2\}\} \in P(A)$ ✓ Correct
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. $2$ ✓ Correct
B. $1$
C. $4$
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. $4$ ✓ Correct
C. $2$
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. $8$
B. $3$
C. $6$
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. $\{1, 2\} \in P(A)$
B. $1 \in P(A)$ ✓ Correct
C. $\{1\} \in P(A)$
D. $\emptyset \in P(A)$
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 · hard · numerical
If $n(A) = m$, the number of subsets of $A$ having at least $2$ elements is:
A. $2^m - m$
B. $2^m - 1$
C. $2^m - m - 1$ ✓ Correct
D. $2^{m-1}$
Solution: Total subsets $2^m$; subtract those with 0 elements (1, the empty set) and 1 element ($m$): $2^m - 1 - m$.
Q11 — Power Set · hard · numerical
A set $A$ has $n$ elements. The number of subsets of $A$ with an even number of elements is:
A. $2^{n-1}$ ✓ Correct
B. $2^n$
C. $2^{n-1} - 1$
D. $n^2$
Solution: For $n \ge 1$, exactly half of the $2^n$ subsets have an even size: $2^{n-1}$.
Q12 — Power Set · hard · theory
If $A = \{\emptyset, \{\emptyset\}\}$, then the number of elements in $P(A)$ is:
A. $8$
B. $4$ ✓ Correct
C. $2$
D. $1$
Solution: $A$ has 2 elements ($\emptyset$ and $\{\emptyset\}$), so $n(P(A)) = 2^2 = 4$.
Q13 — Power Set · hard · numerical
If $n(P(A)) = 4\,n(P(B))$ and $n(A) + n(B) = 6$, then $n(A)$ equals:
A. $5$
B. $3$
C. $2$
D. $4$ ✓ Correct
Solution: $2^{n(A)} = 4\cdot2^{n(B)} = 2^{n(B)+2} \Rightarrow n(A) = n(B)+2$; with $n(A)+n(B)=6$, we get $n(B)=2,\ n(A)=4$.