If \(A\setminus C=B\setminus C\) and \(A\cap C=B\cap C\), what is the conclusion about \(A\) and \(B\)?
Answer and explanation
Correct answer: \(A=B\)
Every set can be decomposed into two disjoint parts relative to \(C\): the elements outside \(C\), namely \(A\setminus C\), and the elements inside \(C\), namely \(A\cap C\). The two corresponding parts of \(A\) and \(B\) are given equal. Their unions therefore are equal: \(A=(A\setminus C)\cup(A\cap C)=(B\setminus C)\cup(B\cap C)=B\). Hence option A must hold.
Frequently asked questions
What is the correct answer to this question?
\(A=B\)
Why is this the correct answer?
Every set can be decomposed into two disjoint parts relative to \(C\): the elements outside \(C\), namely \(A\setminus C\), and the elements inside \(C\), namely \(A\cap C\). The two corresponding parts of \(A\) and \(B\) are given equal. Their unions therefore are equal: \(A=(A\setminus C)\cup(A\cap C)=(B\setminus C)\cup(B\cap C)=B\). Hence option A must hold.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Operations on Sets (Union, Intersection, Difference).
