If A ⊆ U, |U| = 9, and |A| = 4, what is the number of proper subsets of P(A')?
Answer and explanation
Correct answer: 4294967295
Because A has 4 elements and U has 9 elements, its complement A' has 9 − 4 = 5 elements. Therefore P(A') has 2^5 = 32 elements. A set with 32 elements has 2^32 total subsets, including itself. Proper subsets exclude the set itself, so their number is 2^32 − 1 = 4,294,967,295. Thus option D is correct. The value 1023 would incorrectly correspond to proper subsets of a 10-element set, not a 32-element set.
Frequently asked questions
What is the correct answer to this question?
4294967295
Why is this the correct answer?
Because A has 4 elements and U has 9 elements, its complement A' has 9 − 4 = 5 elements. Therefore P(A') has 2^5 = 32 elements. A set with 32 elements has 2^32 total subsets, including itself. Proper subsets exclude the set itself, so their number is 2^32 − 1 = 4,294,967,295. Thus option D is correct. The value 1023 would incorrectly correspond to proper subsets of a 10-element set, not a 32-element set.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Power Set and Subsets.
