De Morgan's Laws (Set Theory)/Relative Complement/General Case/Complement of Union

Theorem

Let $S$ be a set.

Let $T$ be a subset of $S$.

Let $\mathcal P \left({T}\right)$ be the power set of $T$.

Let $\mathbb T \subseteq \mathcal P \left({T}\right)$.

Then:

$\displaystyle \complement_S \left({\bigcup \mathbb T}\right) = \bigcap_{H \mathop \in \mathbb T} \complement_S \left({H}\right)$

$\displaystyle \complement_S \left({\bigcup \mathbb T}\right)$

$=$

$\displaystyle S \setminus \left({\bigcup \mathbb T}\right)$

$\displaystyle $

$=$

$\displaystyle \bigcap_{H \mathop \in \mathbb T} \left({S \setminus H}\right)$

$\displaystyle $

$=$

$\displaystyle \bigcap_{H \mathop \in \mathbb T} \complement_S \left({H}\right)$

$\blacksquare$

This entry was named for Augustus De Morgan.

1965: Seth Warner: Modern Algebra ... (previous) ... (next): Exercise $3.6 \ \text{(d)}$