# De Morgan's Laws (Set Theory)/Relative Complement/Family of Sets/Complement of Union

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## Theorem

Let $S$ be a set.

Let $\family {S_i}_{i \mathop \in I}$ be a family of subsets of $S$.

Then:

- $\ds \relcomp S {\bigcup_{i \mathop \in I} S_i} = \bigcap_{i \mathop \in I} \relcomp S {S_i}$

## Proof

\(\ds \relcomp S {\bigcup_{i \mathop \in I} S_i}\) | \(=\) | \(\ds S \setminus \paren {\bigcup_{i \mathop \in I} S_i}\) | Definition of Relative Complement | |||||||||||

\(\ds \) | \(=\) | \(\ds \bigcap_{i \mathop \in I} \paren {S \setminus S_i}\) | De Morgan's Laws for Set Difference: Difference with Union | |||||||||||

\(\ds \) | \(=\) | \(\ds \bigcap_{i \mathop \in I} \relcomp S {S_i}\) | 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): $2.22$