De Morgan's Laws (Set Theory)/Set Complement/General Case/Complement of Union
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Theorem
Let $\mathbb T$ be a set of sets, all of which are subsets of a universe $\mathbb U$.
Then:
- $\ds \map \complement {\bigcup \mathbb T} = \bigcap_{H \mathop \in \mathbb T} \map \complement H$
Proof
\(\ds \map \complement {\bigcup \mathbb T}\) | \(=\) | \(\ds \mathbb U \setminus \paren {\bigcup \mathbb T}\) | Definition of Set Complement | |||||||||||
\(\ds \) | \(=\) | \(\ds \bigcap_{H \mathop \in \mathbb T} \paren {\mathbb U \setminus H}\) | De Morgan's Laws for Set Difference: Difference with Union | |||||||||||
\(\ds \) | \(=\) | \(\ds \bigcap_{H \mathop \in \mathbb T} \map \complement H\) | Definition of Set Complement |
$\blacksquare$
Source of Name
This entry was named for Augustus De Morgan.
Sources
- 1964: Steven A. Gaal: Point Set Topology ... (previous) ... (next): Introduction to Set Theory: $1$. Elementary Operations on Sets
- 1971: Robert H. Kasriel: Undergraduate Topology ... (previous) ... (next): $\S 1.8$: Collections of Sets: Exercise $5$
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 7.1 \ \text{(ii)}$: Unions and Intersections