De Morgan's Laws (Set Theory)/Set Difference/General Case
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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.
Sources
- 2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 1$: Fundamental Concepts: Exercise $9$