For three sets A, B and C, show that


For three sets AB and C, show that

(i) $A \cap B=A \cap C$ need not imply $B=C$.

(ii) $A \subset B \Rightarrow C-B \subset C-A$


(i) Let A = {2, 4, 5, 6},  B = {6, 7, 8, 9} and C = {6, 10, 11, 12,13}

So, $A \cap B=\{6\}$ and $A \cap C=\{6\}$

Hence, $A \cap B=A \cap C$ but $B \neq C$


Let $z \in C-B \quad \ldots(1)$

$\Rightarrow z \in C$ and $z \notin B$

$\Rightarrow z \in C$ and $z \notin A \quad[\because A \subset B]$

$\Rightarrow z \in C-A$    ...(2)

From $(1)$ and $(2)$, we get

$C-B \subset C-A$

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