Answer
$for\,\,all\,\,sets\,\,A,B\\
if\,\,A\subseteq B\,\,then\,\,B^{c}\subseteq A^{c}\\
x\in B^{c}\Rightarrow x\notin B \\
if\,\,x\in A \Rightarrow x\in B (because\,\,A\subseteq B)\,\,\\
but\,\,x\notin B\,\,so\,\,x\notin A\Rightarrow x\in A^{c}\\
\because x\in B^{c}\Rightarrow x\in A^{c}\\
\therefore B^{c}\subseteq A^{c}$
Work Step by Step
$for\,\,all\,\,sets\,\,A,B\\
if\,\,A\subseteq B\,\,then\,\,B^{c}\subseteq A^{c}\\
proof:\,\,\\x\in B^{c}\Rightarrow x\notin B \\
if\,\,x\in A \Rightarrow x\in B (because\,\,A\subseteq B)\,\,\\
but\,\,x\notin B\,\,so\,\,x\notin A\Rightarrow x\in A^{c}\\
\because x\in B^{c}\Rightarrow x\in A^{c}\\
\therefore B^{c}\subseteq A^{c}$
