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

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): $2.22$