If U = {1, 2, 3, 4, 5, 6, 7, 8, 9} and A = {x ∈ U : x is prime}, how many elements are in P(A′)?
Answer and explanation
Correct answer: 32
The prime numbers in U are 2, 3, 5, and 7, so A = {2, 3, 5, 7}. Notice that 1 is not prime. The complement is therefore A′ = U − A = {1, 4, 6, 8, 9}, containing 5 elements. The power set of a finite set with n elements contains 2ⁿ subsets. Thus n(P(A′)) = 2⁵ = 32. Therefore, option B is correct. The main steps are identifying the primes, forming the complement, and then applying the power-set formula.
Frequently asked questions
What is the correct answer to this question?
32
Why is this the correct answer?
The prime numbers in U are 2, 3, 5, and 7, so A = {2, 3, 5, 7}. Notice that 1 is not prime. The complement is therefore A′ = U − A = {1, 4, 6, 8, 9}, containing 5 elements. The power set of a finite set with n elements contains 2ⁿ subsets. Thus n(P(A′)) = 2⁵ = 32. Therefore, option B is correct. The main steps are identifying the primes, forming the complement, and then applying the power-set formula.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Power Set and Subsets.
