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

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Theorem

Let $\family {S_i}_{i \mathop \in I}$ be a family of sets, all of which are subsets of a universe $\mathbb U$.


Then:

$\displaystyle \map \complement {\bigcap_{i \mathop \in I} S_i} = \bigcup_{i \mathop \in I} \map \complement {S_i}$


Proof

\(\displaystyle \map \complement {\bigcap_{i \mathop \in I} S_i}\) \(=\) \(\displaystyle \mathbb U \setminus \paren {\bigcap_{i \mathop \in I} S_i}\) Definition of Set Complement
\(\displaystyle \) \(=\) \(\displaystyle \bigcup_{i \mathop \in I} \paren {\mathbb U \setminus S_i}\) De Morgan's Laws for Set Difference: Difference with Intersection
\(\displaystyle \) \(=\) \(\displaystyle \bigcup_{i \mathop \in I} \map \complement {S_i}\) Definition of Set Complement

$\blacksquare$


Also see


Source of Name

This entry was named for Augustus De Morgan.


Sources