Question:
For any two sets of A and B, prove that:
(i) $A^{\prime} \cup B=U \Rightarrow A \subset B$
(ii) $B^{\prime} \subset A^{\prime} \Rightarrow A \subset B$
Solution:
(i) Let $a \in A$.
$\Rightarrow a \in \mathrm{U}$
$\Rightarrow a \in A^{\prime} \cup B \quad\left(\because U=A^{\prime} \cup B\right)$
$\Rightarrow a \in B \quad\left(\because a \notin A^{\prime}\right)$
Hence, $A \subset B$
(ii) Let $a \in A$.
$\Rightarrow a \notin A^{\prime}$
$\Rightarrow a \notin B^{\prime} \quad\left(\because B^{\prime} \subset A^{\prime}\right)$
$\Rightarrow a \in B$
Hence, $A \subset B$