If A ∪ B = A ∩ C, which conclusion must be true?
Answer and explanation
Correct answer: B ⊆ A and A ⊆ C
Use the basic containment properties of union and intersection. Since A ∩ C is always a subset of A, the equality A ∪ B = A ∩ C gives A ∪ B ⊆ A. Because B ⊆ A ∪ B, it follows that B ⊆ A. Also A ⊆ A ∪ B = A ∩ C, so every element of A lies in C; hence A ⊆ C. Therefore option A is necessary. The other choices require relationships not forced by the equality.
Frequently asked questions
What is the correct answer to this question?
B ⊆ A and A ⊆ C
Why is this the correct answer?
Use the basic containment properties of union and intersection. Since A ∩ C is always a subset of A, the equality A ∪ B = A ∩ C gives A ∪ B ⊆ A. Because B ⊆ A ∪ B, it follows that B ⊆ A. Also A ⊆ A ∪ B = A ∩ C, so every element of A lies in C; hence A ⊆ C. Therefore option A is necessary. The other choices require relationships not forced by the 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).
