For any two sets of A and B, prove that:

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, $A \subset B$

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