If \(P(A)\) has 16 elements, how many elements does A have?
Answer and explanation
Correct answer: 4
If a finite set A has n elements, then each element has two choices when forming a subset: it is either included or excluded. Therefore the power set has \(|P(A)|=2^n\) elements. Here \(2^n=16=2^4\), so n=4. Thus A contains 4 elements. The number 16 describes the total number of subsets, not the number of original elements. For comparison, sets with 2 and 3 elements have power sets of sizes 4 and 8 respectively.
Frequently asked questions
What is the correct answer to this question?
4
Why is this the correct answer?
If a finite set A has n elements, then each element has two choices when forming a subset: it is either included or excluded. Therefore the power set has \(|P(A)|=2^n\) elements. Here \(2^n=16=2^4\), so n=4. Thus A contains 4 elements. The number 16 describes the total number of subsets, not the number of original elements. For comparison, sets with 2 and 3 elements have power sets of sizes 4 and 8 respectively.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Power Set and Subsets.
