If A ∩ B = A ∪ B ∪ C, which conclusion about C must be true?
Answer and explanation
Correct answer: C ⊆ A ∩ B and A = B
For all sets, A ∩ B ⊆ A ⊆ A ∪ B. Therefore the equality A ∩ B = A ∪ B ∪ C implies that A ∪ B is also contained in A ∩ B. Since A ∩ B is always contained in A ∪ B, we obtain A ∩ B = A ∪ B, which is possible exactly when A = B. The equality also forces every element of C to lie in A ∩ B, so C ⊆ A ∩ B.
Frequently asked questions
What is the correct answer to this question?
C ⊆ A ∩ B and A = B
Why is this the correct answer?
For all sets, A ∩ B ⊆ A ⊆ A ∪ B. Therefore the equality A ∩ B = A ∪ B ∪ C implies that A ∪ B is also contained in A ∩ B. Since A ∩ B is always contained in A ∪ B, we obtain A ∩ B = A ∪ B, which is possible exactly when A = B. The equality also forces every element of C to lie in A ∩ B, so C ⊆ A ∩ B.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Operations on Sets (Union, Intersection, Difference).
