If \(U=\{0,1,2,3,4,5\}\) and \(A=\{x:x^2=x\}\), what is \(n(\mathcal{P}(A'))\)? Here, \(A'\) denotes the complement of \(A\) in \(U\).
Answer and explanation
Correct answer: 16
Solve the defining equation: \(x^2=x\) implies \(x^2-x=0\), or \(x(x-1)=0\). Hence \(x=0\) or \(x=1\), so \(A=\{0,1\}\). Its complement in \(U\) is \(A'=\{2,3,4,5\}\), which has four elements. A set with \(k\) elements has \(2^k\) subsets, so \(n(\mathcal{P}(A'))=2^4=16\). Therefore option A is correct; 4 is only the size of \(A'\), not of its power set.
Frequently asked questions
What is the correct answer to this question?
16
Why is this the correct answer?
Solve the defining equation: \(x^2=x\) implies \(x^2-x=0\), or \(x(x-1)=0\). Hence \(x=0\) or \(x=1\), so \(A=\{0,1\}\). Its complement in \(U\) is \(A'=\{2,3,4,5\}\), which has four elements. A set with \(k\) elements has \(2^k\) subsets, so \(n(\mathcal{P}(A'))=2^4=16\). Therefore option A is correct; 4 is only the size of \(A'\), not of its power set.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Power Set and Subsets.
