Complement of unions mh - Rhea

Theorem

Let $\{E_{\alpha}\}$ be a (finite or infinite) collection of sets $E_{\alpha}$ . Then,

(\bigcup_{\alpha}E_{\alpha})^C = \bigcap_{\alpha}(E_{\alpha}^C)

\begin{align} x\in (\bigcup_{\alpha}E_{\alpha})^C &\Rightarrow x \notin \bigcup_{\alpha}E_{\alpha} \\ &\Rightarrow x \notin E_{\alpha} \;\forall \alpha \\ &\Rightarrow x \in E_{\alpha}^C \;\forall\alpha \\ &\Rightarrow x \in\bigcap E_{\alpha}^C \\ &\Rightarrow (\bigcup_{\alpha}E_{\alpha})^C \subset \bigcap_{\alpha}(E_{\alpha}^C) \end{align}

Conversely,

\begin{align} x \in \bigcap_{\alpha}(E_{\alpha}^C) &\Rightarrow x \in E_{\alpha}^C\;\forall\alpha \\ &\Rightarrow x \notin E_{\alpha}\; \forall \alpha \\ &\Rightarrow x \notin\bigcup_{\alpha}E_{\alpha} \\ &\Rightarrow x \in \bigcup_{\alpha}E_{\alpha})^C \\ &\Rightarrow \bigcap_{\alpha}(E_{\alpha}^C)\subset (\bigcup_{\alpha}E_{\alpha})^C \end{align}

Therefore,

(\bigcup_{\alpha}E_{\alpha})^C = \bigcap_{\alpha}(E_{\alpha}^C)

$\blacksquare$