Power Set and Its Elements

easyIf A = {5, 10}, find n(P(A)).

easyIf A = {x}, find the number of elements in P(A).

mediumIf A = {1, 2, 3}, how many elements are in P(A)?

mediumIf n(A) = 4, find n(P(A)).

hardIf n(P(A)) = 64, find n(A).

hardIf A = {∅, {1}}, find n(P(A)).

very hardIf n(P(A)) = 256, find n(P(P(A))).

Assertion (A): For A = {1, 2}, the power set P(A) contains 4 elements.
Reason (R): The number of elements in a power set is given by 2ⁿ where n is the number of elements in the set.

Choose the correct option:
(A) Both A and R are true and R is the correct explanation of A.
(B) Both A and R are true but R is not the correct explanation of A.
(C) A is true but R is false.
(D) A is false but R is true.

Assertion (A): If A = {1, 2, 3}, then 3 ∈ P(A).
Reason (R): The elements of P(A) are subsets of A, and {3} is a subset of A, not 3 itself.

Choose the correct option:
(A) Both A and R are true and R is the correct explanation of A.
(B) Both A and R are true but R is not the correct explanation of A.
(C) A is true but R is false.
(D) A is false but R is true.

mediumGiven A = {a, b, c}, find: (i) n(A) (ii) n(P(A)) (iii) List all elements of P(A).

mediumGiven A = {1, 2}, find: (i) P(A) (ii) n(P(P(A))) (iii) Is {1} ∈ P(A)?

hardGiven A = {0, 1, 2}, find: (i) n(P(A)) (ii) Number of proper subsets of A (iii) Number of non-empty subsets of A.

hardLet A = {1, 2, 3, 4}. Calculate the following: (i) n(P(A)) (ii) Number of subsets containing 1 (iii) Number of subsets not containing 1 (iv) Number of proper subsets (v) Verify n(P(A)) = (ii) + (iii).

very hardIf A = {a, b, c, d, e}, find: (i) n(P(A)) (ii) Number of subsets with at least one element (iii) Number of subsets with exactly 2 elements (iv) Number of subsets with at most 2 elements (v) Sum of (iii) and (iv).

Case Study: A library has a special collection of 4 rare books labeled {B₁, B₂, B₃, B₄}. A researcher wants to study all possible combinations of these books.

If A = {1, 2, 3}, find the number of elements in P(A).

If n(A) = 5, find the number of proper subsets of A.

If A = {a, b}, find P(A).

If n(P(A)) = 16, find n(A).