If \(A\subseteq B\), which statement about their complements is true?
Answer and explanation
Correct answer: \(B^c\subseteq A^c\)
Complements reverse the direction of inclusion. Since \(A\subseteq B\), every element outside \(B\) is certainly outside \(A\) as well. Thus, if \(x\in B^c\), then \(x\notin B\), which implies \(x\notin A\), so \(x\in A^c\). Therefore \(B^c\subseteq A^c\), making option A correct. Option B reverses this result, while option C requires \(A=B\). Option D is not generally true; it would impose an additional condition on the universal set.
Frequently asked questions
What is the correct answer to this question?
\(B^c\subseteq A^c\)
Why is this the correct answer?
Complements reverse the direction of inclusion. Since \(A\subseteq B\), every element outside \(B\) is certainly outside \(A\) as well. Thus, if \(x\in B^c\), then \(x\notin B\), which implies \(x\notin A\), so \(x\in A^c\). Therefore \(B^c\subseteq A^c\), making option A correct. Option B reverses this result, while option C requires \(A=B\). Option D is not generally true; it would impose an additional condition on the universal set.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Complement of a Set and Its Properties.
