If A \ C = B \ C, which statement is always true?
Answer and explanation
Correct answer: A \ (B ∪ C) = ∅ and B \ (A ∪ C) = ∅
The equality A \ C = B \ C says that A and B contain exactly the same elements outside C. Thus any element of A that is not in B cannot be outside C; otherwise it would appear in A \ C but not in B \ C. Therefore every element of A lies in B ∪ C, which is equivalent to A \ (B ∪ C) = ∅. By the same reasoning, every element of B lies in A ∪ C, so B \ (A ∪ C) = ∅. Hence option A is always true.
Frequently asked questions
What is the correct answer to this question?
A \ (B ∪ C) = ∅ and B \ (A ∪ C) = ∅
Why is this the correct answer?
The equality A \ C = B \ C says that A and B contain exactly the same elements outside C. Thus any element of A that is not in B cannot be outside C; otherwise it would appear in A \ C but not in B \ C. Therefore every element of A lies in B ∪ C, which is equivalent to A \ (B ∪ C) = ∅. By the same reasoning, every element of B lies in A ∪ C, so B \ (A ∪ C) = ∅. Hence option A is always true.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Operations on Sets (Union, Intersection, Difference).
