If the power set P(A) has 64 elements, how many proper subsets does A have?
Answer and explanation
Correct answer: 63
If A has n elements, then its power set has 2ⁿ elements. Given |P(A)| = 64, we have 2ⁿ = 64 = 2⁶, so n = 6. A set with six elements has 2⁶ = 64 total subsets. Exactly one of these subsets is A itself, and a proper subset must be different from A. Therefore the number of proper subsets is 64 − 1 = 63. The empty set is included among the proper subsets.
Frequently asked questions
What is the correct answer to this question?
63
Why is this the correct answer?
If A has n elements, then its power set has 2ⁿ elements. Given |P(A)| = 64, we have 2ⁿ = 64 = 2⁶, so n = 6. A set with six elements has 2⁶ = 64 total subsets. Exactly one of these subsets is A itself, and a proper subset must be different from A. Therefore the number of proper subsets is 64 − 1 = 63. The empty set is included among the proper subsets.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Power Set and Subsets.
