Class JEE Mathematics Sets, Relations, and Functions Q #1073
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4 Marks 2009 AIEEE MCQ SINGLE
If $A$, $B$ and $C$ are three sets such that $A \cap B = A \cap C$ and $A \cup B = A \cup C$, then :
(A) $A = C$
(B) $B = C$
(C) $A \cap B = \phi$
(D) $A = B$
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Correct Answer: B
Explanation
Given $A \cap B = A \cap C$ and $A \cup B = A \cup C$. From the principle of inclusion-exclusion: $|A \cup B| = |A| + |B| - |A \cap B|$ and $|A \cup C| = |A| + |C| - |A \cap C|$. Since $A \cup B = A \cup C$ and $A \cap B = A \cap C$, then $|A \cup B| = |A \cup C|$ and $|A \cap B| = |A \cap C|$. Thus, $|A| + |B| - |A \cap B| = |A| + |C| - |A \cap C|$. This simplifies to $|B| = |C|$. Since $A \cap B = A \cap C$ and $A \cup B = A \cup C$, it means that $B$ and $C$ contain the same elements and therefore are equal to each other ($B=C$).

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