Question

For any two sets of A and B, prove that:
(i) $A\text{'}\cup B=U\Rightarrow A\subset B$
(ii) $B\text{'}\subset A\text{'}\Rightarrow A\subset B$

Answer 1

$$\begin{gathered} \left(\mathrm{i}\right)\mathrm{Let}a\in A. \\ \Rightarrow a\in \mathrm{U} \\ \Rightarrow a\in A\text{'}\cup B\left(\because U=A\text{'}\cup B\right) \\ \Rightarrow a\in B\left(\because a\notin A\text{'}\right) \\ \mathrm{Hence},A\subset B. \end{gathered}$$

$$\begin{gathered} \left(\mathrm{ii}\right)\mathrm{Let}a\in A. \\ \Rightarrow a\notin A\text{'} \\ \Rightarrow a\notin B\text{'}\left(\because B\text{'}\subset A\text{'}\right) \\ \Rightarrow a\in B \\ \mathrm{Hence},A\subset B. \end{gathered}$$