De Morgan's Laws (Set Theory)/Relative Complement/General Case/Complement of Union
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Contents
Theorem
Let $S$ be a set.
Let $T$ be a subset of $S$.
Let $\mathcal P \left({T}\right)$ be the power set of $T$.
Let $\mathbb T \subseteq \mathcal P \left({T}\right)$.
Then:
- $\displaystyle \complement_S \left({\bigcup \mathbb T}\right) = \bigcap_{H \mathop \in \mathbb T} \complement_S \left({H}\right)$
Proof
\(\displaystyle \complement_S \left({\bigcup \mathbb T}\right)\) | \(=\) | \(\displaystyle S \setminus \left({\bigcup \mathbb T}\right)\) | Definition of Relative Complement | ||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle \bigcap_{H \mathop \in \mathbb T} \left({S \setminus H}\right)\) | De Morgan's Laws: Difference with Union | ||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle \bigcap_{H \mathop \in \mathbb T} \complement_S \left({H}\right)\) | Definition of Relative Complement |
$\blacksquare$
Source of Name
This entry was named for Augustus De Morgan.
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Exercise $3.6 \ \text{(d)}$