De Morgan's Laws (Set Theory)/Set Complement/Family of Sets/Complement of Intersection
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Theorem
Let $\family {S_i}_{i \mathop \in I}$ be a family of sets, all of which are subsets of a universal set $\mathbb U$.
Then:
- $\ds \map \complement {\bigcap_{i \mathop \in I} S_i} = \bigcup_{i \mathop \in I} \map \complement {S_i}$
Proof
| \(\ds \map \complement {\bigcap_{i \mathop \in I} S_i}\) | \(=\) | \(\ds \mathbb U \setminus \paren {\bigcap_{i \mathop \in I} S_i}\) | Definition of Set Complement | |||||||||||
| \(\ds \) | \(=\) | \(\ds \bigcup_{i \mathop \in I} \paren {\mathbb U \setminus S_i}\) | De Morgan's Laws for Set Difference: Difference with Intersection | |||||||||||
| \(\ds \) | \(=\) | \(\ds \bigcup_{i \mathop \in I} \map \complement {S_i}\) | Definition of Set Complement |
$\blacksquare$
Also see
Source of Name
This entry was named for Augustus De Morgan.
Sources
- 1970: Avner Friedman: Foundations of Modern Analysis ... (previous) ... (next): $\S 1.1$: Rings and Algebras
- 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $1$: Theory of Sets: $\S 4$: Indexed Families of Sets: Theorem $4.2$