If A ∪ B = A ∩ C, which conclusion must be true?
Answer and explanation
Correct answer: B ⊆ A and A ⊆ C
Let the common set be S, where S = A ∪ B = A ∩ C. Because A ⊆ A ∪ B, we have A ⊆ S. Also, A ∩ C ⊆ A, so S ⊆ A. Therefore S = A. From A ∪ B = A, every element of B must already belong to A, giving B ⊆ A. From A ∩ C = A, every element of A must belong to C, giving A ⊆ C. Hence option A is the only definite conclusion; the other options need not hold.
Frequently asked questions
What is the correct answer to this question?
B ⊆ A and A ⊆ C
Why is this the correct answer?
Let the common set be S, where S = A ∪ B = A ∩ C. Because A ⊆ A ∪ B, we have A ⊆ S. Also, A ∩ C ⊆ A, so S ⊆ A. Therefore S = A. From A ∪ B = A, every element of B must already belong to A, giving B ⊆ A. From A ∩ C = A, every element of A must belong to C, giving A ⊆ C. Hence option A is the only definite conclusion; the other options need not 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).
