De Morgan's Laws (Set Theory)/Relative Complement/Family of Sets/Complement of Union
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Contents
Theorem
Let $S$ be a set.
Let $\family {S_i}_{i \mathop \in I}$ be a family of subsets of $S$.
Then:
- $\displaystyle \relcomp S {\bigcup_{i \mathop \in I} \mathbb S_i} = \bigcap_{i \mathop \in I} \relcomp S {S_i}$
Proof
| \(\displaystyle \relcomp S {\bigcup_{i \mathop \in I} S_i}\) | \(=\) | \(\displaystyle S \setminus \paren {\bigcup_{i \mathop \in I} S_i}\) | Definition of Relative Complement | ||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \bigcap_{i \mathop \in I} \paren {S \setminus S_i}\) | De Morgan's Laws for Set Difference: Difference with Union | ||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \bigcap_{i \mathop \in I} \relcomp S {S_i}\) | Definition of Relative Complement |
$\blacksquare$
Source of Name
This entry was named for Augustus De Morgan.
Sources
- 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 6$. Indexed families; partitions; equivalence relations: Exercise $2$
- 1993: Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (2nd ed.) ... (previous) ... (next): $\S 1$: Naive Set Theory: $\S 1.4$: Sets of Sets: Exercise $1.4.5 \ \text{(i)}$
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- 1953: Walter Rudin: Principles of Mathematical Analysis ... (next): $2.22$
