If A ⊆ U, |A| = m, and |U| = 2m, when does |P(A)| = |P(A')| hold?
Advertisement
Answer and explanation
Correct answer: Always
Since A is a subset of U, its complement has cardinality |A'| = |U| − |A| = 2m − m = m. The power set of any set with m elements has 2^m elements. Therefore |P(A)| = 2^m and |P(A')| = 2^m, so the two cardinalities are equal for every allowed value of m, not just for a special value. Hence option A is correct.
Frequently asked questions
What is the correct answer to this question?
Always
Why is this the correct answer?
Since A is a subset of U, its complement has cardinality |A'| = |U| − |A| = 2m − m = m. The power set of any set with m elements has 2^m elements. Therefore |P(A)| = 2^m and |P(A')| = 2^m, so the two cardinalities are equal for every allowed value of m, not just for a special value. Hence option A is correct.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Power Set and Subsets.
Advertisement