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

Theorem

Let $S$ and $T$ be sets.

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

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

Then:

Difference with Intersection

$\ds S \setminus \bigcap \mathbb T = \bigcup_{T' \mathop \in \mathbb T} \paren {S \setminus T'}$

where:

$\ds \bigcap \mathbb T := \set {x: \forall T' \in \mathbb T: x \in T'}$

that is, the intersection of $\mathbb T$

Difference with Union

$\ds S \setminus \bigcup \mathbb T = \bigcap_{T' \mathop \in \mathbb T} \paren {S \setminus T'}$

where:

$\ds \bigcup \mathbb T := \set {x: \exists T' \in \mathbb T: x \in T'}$

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

Source of Name

This entry was named for Augustus De Morgan.