De Morgan's Laws (Set Theory)/Relative Complement/General Case/Complement of Union
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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:
- $\ds \relcomp S {\bigcup \mathbb T} = \bigcap_{H \mathop \in \mathbb T} \relcomp S H$
Proof
\(\ds \relcomp S {\bigcup \mathbb T}\) | \(=\) | \(\ds S \setminus \paren {\bigcup \mathbb T}\) | Definition of Relative Complement | |||||||||||
\(\ds \) | \(=\) | \(\ds \bigcap_{H \mathop \in \mathbb T} \paren {S \setminus H}\) | De Morgan's Laws: Difference with Union | |||||||||||
\(\ds \) | \(=\) | \(\ds \bigcap_{H \mathop \in \mathbb T} \relcomp S H\) | Definition of Relative Complement |
$\blacksquare$
Source of Name
This entry was named for Augustus De Morgan.
Sources
- 1951: J.C. Burkill: The Lebesgue Integral ... (previous) ... (next): Chapter $\text {I}$: Sets of Points: $1 \cdot 1$. The algebra of sets
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 3$: Unions and Intersections of Sets: Exercise $3.6 \ \text{(d)}$