If \(A\cup(B\cap C)=A\), which inclusion must be true?
Answer and explanation
Correct answer: \(B\cap C\subseteq A\)
For any sets \(X\) and \(A\), the equality \(A\cup X=A\) holds exactly when every element of \(X\) is already in \(A\), that is, \(X\subseteq A\). Here \(X=B\cap C\), so the required conclusion is \(B\cap C\subseteq A\). The equality does not require either \(B\) or \(C\) individually to be contained in \(A\).
Frequently asked questions
What is the correct answer to this question?
\(B\cap C\subseteq A\)
Why is this the correct answer?
For any sets \(X\) and \(A\), the equality \(A\cup X=A\) holds exactly when every element of \(X\) is already in \(A\), that is, \(X\subseteq A\). Here \(X=B\cap C\), so the required conclusion is \(B\cap C\subseteq A\). The equality does not require either \(B\) or \(C\) individually to be contained in \(A\).
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Operations on Sets (Union, Intersection, Difference).
