Question:
For three sets A, B and C, show that
(i) $A \cap B=A \cap C$ need not imply $B=C$.
(ii) $A \subset B \Rightarrow C-B \subset C-A$
Solution:
(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 $B \neq C$
(ii)
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
$C-B \subset C-A$