If \(A\subseteq B\), what is the relation between the power sets \(\mathcal{P}(A)\) and \(\mathcal{P}(B)\)?
Answer and explanation
Correct answer: \(\mathcal{P}(A)\subseteq\mathcal{P}(B)\)
Every member of \(\mathcal{P}(A)\) is a subset of \(A\). Since \(A\subseteq B\), any subset of \(A\) is automatically also a subset of \(B\). Therefore, every element of \(\mathcal{P}(A)\) belongs to \(\mathcal{P}(B)\), giving \(\mathcal{P}(A)\subseteq\mathcal{P}(B)\). Equality occurs only when \(A=B\); in general, \(B\) may have additional elements and therefore additional subsets.
Frequently asked questions
What is the correct answer to this question?
\(\mathcal{P}(A)\subseteq\mathcal{P}(B)\)
Why is this the correct answer?
Every member of \(\mathcal{P}(A)\) is a subset of \(A\). Since \(A\subseteq B\), any subset of \(A\) is automatically also a subset of \(B\). Therefore, every element of \(\mathcal{P}(A)\) belongs to \(\mathcal{P}(B)\), giving \(\mathcal{P}(A)\subseteq\mathcal{P}(B)\). Equality occurs only when \(A=B\); in general, \(B\) may have additional elements and therefore additional subsets.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Power Set and Subsets.
