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

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Solution

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Let $A = {0, 1}$
$B = {0, 2, 3}$
$C = {0, 4, 5}$
So, $A \cap B = {0}$
and $A \cap C = {0}$
Here, $A \cap B = A \cap C = {0}$
However, $B \neq C$ as $2 \in B$ and $2 \notin C$

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