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

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Theorem

Let $S$ be a set.

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

Let $\powerset T$ be the power set of $T$.

Let $\mathbb T \subseteq \powerset T$.


Then:

$\displaystyle \relcomp S {\bigcap \mathbb T} = \bigcup_{H \mathop \in \mathbb T} \relcomp S H$


Proof

\(\displaystyle \relcomp S {\bigcap \mathbb T}\) \(=\) \(\displaystyle S \setminus \paren {\bigcap \mathbb T}\) Definition of Relative Complement
\(\displaystyle \) \(=\) \(\displaystyle \bigcup_{H \mathop \in \mathbb T} \paren {S \setminus H}\) De Morgan's Laws: Difference with Intersection
\(\displaystyle \) \(=\) \(\displaystyle \bigcup_{H \mathop \in \mathbb T} \relcomp S H\) Definition of Relative Complement

$\blacksquare$


Source of Name

This entry was named for Augustus De Morgan.


Sources