Q:

Set theory - axiom of powers

I am reading Halmos' Naive Set Theory.
He writes: If $\mathcal C $ is a collection of subsets of a set $ E $ (that is, $\mathcal C $ is a subcollection of power set $ \wp(E) $), then write
$\mathcal  D = \{X \in \wp(E):X^{\complement}\in \mathcal C\} $
If $E = \{1,2\} $ and $\mathcal C = \{\{1\},\{2\}\} $ does $\mathcal C =\mathcal  D $?
Thanks in advance,

A:

The answer is "yes".
$\mathcal D=\{X\in\wp(E)\mid X^{\complement}\in\{\{1\},\{2\}\}\}=\{\{1\}^{\complement},\{2\}^{\complement}\}=\{\{2\},\{1\}\}=\mathcal C$

