If \(A=\{x:x\in\mathbb{N},\ x\leq5\}\) and \(U=\{x:x\in\mathbb{N},\ x\leq8\}\), how many members are in \(\mathcal{P}(U)-\mathcal{P}(A)\)?
Answer and explanation
Correct answer: 224
Using the usual school convention \(\mathbb{N}=\{1,2,3,\ldots\}\), we have \(|A|=5\) and \(|U|=8\). Hence \(|\mathcal{P}(A)|=2^5=32\), while \(|\mathcal{P}(U)|=2^8=256\). Because \(A\subseteq U\), every subset of A is also a subset of U, so \(\mathcal{P}(A)\subseteq\mathcal{P}(U)\). Therefore, \(|\mathcal{P}(U)-\mathcal{P}(A)|=256-32=224\). Option A is correct. This difference is not \(\mathcal{P}(U-A)\).
Frequently asked questions
What is the correct answer to this question?
224
Why is this the correct answer?
Using the usual school convention \(\mathbb{N}=\{1,2,3,\ldots\}\), we have \(|A|=5\) and \(|U|=8\). Hence \(|\mathcal{P}(A)|=2^5=32\), while \(|\mathcal{P}(U)|=2^8=256\). Because \(A\subseteq U\), every subset of A is also a subset of U, so \(\mathcal{P}(A)\subseteq\mathcal{P}(U)\). Therefore, \(|\mathcal{P}(U)-\mathcal{P}(A)|=256-32=224\). Option A is correct. This difference is not \(\mathcal{P}(U-A)\).
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Power Set and Subsets.
