De Morgan's Laws (Set Theory)/Relative Complement/Family of Sets/Complement of Union

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Theorem

Let $S$ be a set.

Let $\family {S_i}_{i \mathop \in I}$ be a family of subsets of $S$.


Then:

$\displaystyle \relcomp S {\bigcup_{i \mathop \in I} \mathbb S_i} = \bigcap_{i \mathop \in I} \relcomp S {S_i}$


Proof

\(\displaystyle \relcomp S {\bigcup_{i \mathop \in I} S_i}\) \(=\) \(\displaystyle S \setminus \paren {\bigcup_{i \mathop \in I} S_i}\) Definition of Relative Complement
\(\displaystyle \) \(=\) \(\displaystyle \bigcap_{i \mathop \in I} \paren {S \setminus S_i}\) De Morgan's Laws for Set Difference: Difference with Union
\(\displaystyle \) \(=\) \(\displaystyle \bigcap_{i \mathop \in I} \relcomp S {S_i}\) Definition of Relative Complement

$\blacksquare$


Source of Name

This entry was named for Augustus De Morgan.


Sources