If \(A\cap B=\varnothing\) and \(A\cup B=A\cup C\), which additional condition is sufficient for \(B\subseteq C\)?
Answer and explanation
Correct answer: \(A\cap C=\varnothing\)
Assume \(A\cap C=\varnothing\) as well. Since \(A\cap B=\varnothing\), every element of \(B\) lies outside \(A\). Equality of the unions \(A\cup B\) and \(A\cup C\) then forces every element of \(B\) to occur in \(C\); otherwise it would appear only in the first union. Hence \(B\subseteq C\), so option A is sufficient.
Frequently asked questions
What is the correct answer to this question?
\(A\cap C=\varnothing\)
Why is this the correct answer?
Assume \(A\cap C=\varnothing\) as well. Since \(A\cap B=\varnothing\), every element of \(B\) lies outside \(A\). Equality of the unions \(A\cup B\) and \(A\cup C\) then forces every element of \(B\) to occur in \(C\); otherwise it would appear only in the first union. Hence \(B\subseteq C\), so option A is sufficient.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Operations on Sets (Union, Intersection, Difference).
