De Morgan's Laws (Set Theory)/Set Difference/General Case

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Theorem

Let $S$ and $T$ be sets.

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

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


Then:

Difference with Intersection

$\displaystyle S \setminus \bigcap \mathbb T = \bigcup_{T' \mathop \in \mathbb T} \left({S \setminus T'}\right)$

where:

$\displaystyle \bigcap \mathbb T := \left\{{x: \forall T' \in \mathbb T: x \in T'}\right\}$

that is, the intersection of $\mathbb T$


Difference with Union

$\displaystyle S \setminus \bigcup \mathbb T = \bigcap_{T' \mathop \in \mathbb T} \left({S \setminus T'}\right)$

where:

$\displaystyle \bigcup \mathbb T := \left\{{x: \exists T' \in \mathbb T: x \in T'}\right\}$

that is, the union of $\mathbb T$.


Source of Name

This entry was named for Augustus De Morgan.


Sources