If \(A\cup B=A\cup C\), which additional statement is sufficient to prove \(B=C\)?
Answer and explanation
Correct answer: \(A\cap B=A\cap C\)
Use the standard decomposition \(B=(B\setminus A)\cup(A\cap B)\). From \(A\cup B=A\cup C\), the parts outside \(A\) are equal: \(B\setminus A=C\setminus A\). The additional condition \(A\cap B=A\cap C\) makes the parts inside \(A\) equal as well. Thus both disjoint components of \(B\) and \(C\) match, so \(B=C\). The other statements do not generally determine equality.
Frequently asked questions
What is the correct answer to this question?
\(A\cap B=A\cap C\)
Why is this the correct answer?
Use the standard decomposition \(B=(B\setminus A)\cup(A\cap B)\). From \(A\cup B=A\cup C\), the parts outside \(A\) are equal: \(B\setminus A=C\setminus A\). The additional condition \(A\cap B=A\cap C\) makes the parts inside \(A\) equal as well. Thus both disjoint components of \(B\) and \(C\) match, so \(B=C\). The other statements do not generally determine equality.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Operations on Sets (Union, Intersection, Difference).
