Sets Operations
If A ∩ B = A ∩ C and A ∪ B = A ∪ C, which of the following must be true?
Select the correct option:
Solution
B = C
- Principle of Uniqueness: If two sets have the same intersection with a third set, and the same union with that third set, they must be identical.
- Proof Sketch: Let x∈B.
- Either x∈A or x∈/A.
- If x∈A: since x∈B, then x∈A∩B. By given, A∩B=A∩C, so x∈A∩C⟹x∈C.
- If x∈/A: since x∈B, then x∈A∪B. By given, A∪B=A∪C. Since x∈A∪C and x∈/A, it must be that x∈C.
- Conclusion: In all cases, x∈B⟹x∈C. Symmetrically, C⊂B. Thus, B=C.
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About This Question
- Subject
- mathematics
- Chapter
- sets, relations and functions
- Topic
- sets operations
- Difficulty
- Hard
- Year
- 2025
Solution
Correct Answer:
B = C
- Principle of Uniqueness: If two sets have the same intersection with a third set, and the same union with that third set, they must be identical.
- Proof Sketch: Let x∈B.
- Either x∈A or x∈/A.
- If x∈A: since x∈B, then x∈A∩B. By given, A∩B=A∩C, so x∈A∩C⟹x∈C.
- If x∈/A: since x∈B, then x∈A∪B. By given, A∪B=A∪C. Since x∈A∪C and x∈/A, it must be that x∈C.
- Conclusion: In all cases, x∈B⟹x∈C. Symmetrically, C⊂B. Thus, B=C.
This hard difficulty mathematics question is from the chapter sets, relations and functions, covering the topic of sets operations. It appeared in the 2025 exam.
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