If \(A=\{1,2,3\}\) and \(B=\{2,3,4,5\}\), how many elements does \(\mathcal{P}(A\cup B)\) contain?
Answer and explanation
Correct answer: 32
First form the union by listing each distinct element only once: \(A\cup B=\{1,2,3,4,5\}\). Thus the union has 5 elements. For any finite set with n elements, its power set contains every possible subset, including the empty set and the full set, and its cardinality is \(2^n\). Hence \(|\mathcal{P}(A\cup B)|=2^5=32\). The repeated elements 2 and 3 are counted only once in the union, so option A is correct.
Frequently asked questions
What is the correct answer to this question?
32
Why is this the correct answer?
First form the union by listing each distinct element only once: \(A\cup B=\{1,2,3,4,5\}\). Thus the union has 5 elements. For any finite set with n elements, its power set contains every possible subset, including the empty set and the full set, and its cardinality is \(2^n\). Hence \(|\mathcal{P}(A\cup B)|=2^5=32\). The repeated elements 2 and 3 are counted only once in the union, so option A is correct.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Operations on Sets (Union, Intersection, Difference).
