Question
Show that $A \cap B = A \cap C$ need not imply B = C.

Answer

Let A = {1, 2, 3, 4} , B = {2, 3, 4, 5, 6}, C = {2, 3, 4, 9, 10}
$\therefore A \cap B$= {1, 2, 3, 4} $ \cap $ {2, 3, 4, 5, 6}
= {2,3, 4}
$A \cap C$= {1, 2, 3, 4}, B = {2, 3, 4, 5, 6}, C = {2, 3, 4, 9, 10}
= {2, 3, 4}
$A \cap C$= {1, 2, 3, 4} $ \cap $ {2, 3, 4, 9, 10}
= {2, 3, 4}
Now we have $A \cap B = A \cap C$
But $B \ne C$

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