If \(A=\{x,y,z,w\}\), how many elements of \(\mathcal{P}(A)\) are disjoint from \(\{x,z\}\)?
Answer and explanation
Correct answer: 4
A subset of \(A\) is disjoint from \(\{x,z\}\) only if it contains neither \(x\) nor \(z\). Therefore, its elements can be chosen only from \(\{y,w\}\). Each of \(y\) and \(w\) may independently be included or excluded, producing \(2^2=4\) subsets: \(\varnothing,\{y\},\{w\},\{y,w\}\). Hence option B is correct; the full power set has 16 elements, but only four are disjoint from the specified set.
Frequently asked questions
What is the correct answer to this question?
4
Why is this the correct answer?
A subset of \(A\) is disjoint from \(\{x,z\}\) only if it contains neither \(x\) nor \(z\). Therefore, its elements can be chosen only from \(\{y,w\}\). Each of \(y\) and \(w\) may independently be included or excluded, producing \(2^2=4\) subsets: \(\varnothing,\{y\},\{w\},\{y,w\}\). Hence option B is correct; the full power set has 16 elements, but only four are disjoint from the specified set.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Power Set and Subsets.
