If A \ (B \ C) = A, which inclusion must be true?
Answer and explanation
Correct answer: A ∩ B ⊆ C
The equality A \ (B \ C) = A means that removing B \ C from A removes no element. Therefore, A has no element in common with B \ C, so A ∩ (B \ C) = ∅. If an element belongs to A ∩ B, it cannot be outside C; otherwise it would belong to A ∩ (B \ C), which is impossible. Hence every element of A ∩ B belongs to C, giving A ∩ B ⊆ C. Option A is correct.
Frequently asked questions
What is the correct answer to this question?
A ∩ B ⊆ C
Why is this the correct answer?
The equality A \ (B \ C) = A means that removing B \ C from A removes no element. Therefore, A has no element in common with B \ C, so A ∩ (B \ C) = ∅. If an element belongs to A ∩ B, it cannot be outside C; otherwise it would belong to A ∩ (B \ C), which is impossible. Hence every element of A ∩ B belongs to C, giving A ∩ B ⊆ C. Option A is correct.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Operations on Sets (Union, Intersection, Difference).
