Maths · Union, intersection and complement of sets and their algebraic properties
Let and be subsets of a set Which one of the following is correct?
Let \( A_{1}, A_{2} \) and \( A_{3} \) be subsets of a set \( X \) Which one of the following is correct?
- A. \( A_{1} \cup A_{2} \cup A_{3} \) is the largest subset of \( X \) containing elements of each of \( A_{1}, A_{2} \) and \( A_{3} \)
- B. \( A_{1} \cup A_{2} \cup A_{3} \) is the smallest subset of \( X \) containing either \( A_{1} \) or \( A_{2} \cup A_{3} \) but not both
- C. The smallest subset of \( X \) containing \( A_{1} \cup A_{2} \) and \( A_{3} \) equals the smallest subset of \( X \) containing both \( A_{1} \) and \( A_{2} \cup A_{3} \) only if \( A_{2}=A_{3} \)
- D. None of these
Step-by-step solution
Option A is false because the union is the smallest superset containing all elements of each A_i, not the largest (which would be X). Option B is false because the union is not the symmetric difference. Option C is false because the smallest subset containing A1∪A2 and A3 is always equal to the smallest subset containing A1 and A2∪A3 (both equal A1∪A2∪A3) regardless of A2=A3. Hence none are correct.
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