If A ⊆ U, |U| = 18, and |P(A′)| = 1024, what is |P(A)|?
Answer and explanation
Correct answer: 2⁸
For every finite set S, the number of elements in its power set is |P(S)| = 2^|S|. Since |P(A′)| = 1024 = 2^10, we obtain |A′| = 10. The complement A′ is taken relative to U, so A and A′ partition U and |A| + |A′| = |U|. Hence |A| = 18 − 10 = 8. Applying the power-set formula again gives |P(A)| = 2^8. Therefore option B is correct. Option C, 2^10, is the size of P(A′), not the size of P(A), which is the key distinction in this problem.
Frequently asked questions
What is the correct answer to this question?
2⁸
Why is this the correct answer?
For every finite set S, the number of elements in its power set is |P(S)| = 2^|S|. Since |P(A′)| = 1024 = 2^10, we obtain |A′| = 10. The complement A′ is taken relative to U, so A and A′ partition U and |A| + |A′| = |U|. Hence |A| = 18 − 10 = 8. Applying the power-set formula again gives |P(A)| = 2^8. Therefore option B is correct. Option C, 2^10, is the size of P(A′), not the size of P(A), which is the key distinction in this problem.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Power Set and Subsets.
