De Morgan's Laws (Set Theory)/Relative Complement/General Case/Complement of Union

Theorem

Let $S$ be a set.

Let $T$ be a subset of $S$.

Let $\powerset T$ be the power set of $T$.

Let $\mathbb T \subseteq \powerset T$.

Then:

$\ds \relcomp S {\bigcup \mathbb T} = \bigcap_{H \mathop \in \mathbb T} \relcomp S H$

Proof

\(\ds \relcomp S {\bigcup \mathbb T}\)

\(=\)

\(\ds S \setminus \paren {\bigcup \mathbb T}\)

Definition of Relative Complement

\(\ds \)

\(=\)

\(\ds \bigcap_{H \mathop \in \mathbb T} \paren {S \setminus H}\)

De Morgan's Laws: Difference with Union

\(\ds \)

\(=\)

\(\ds \bigcap_{H \mathop \in \mathbb T} \relcomp S H\)

Definition of Relative Complement

$\blacksquare$

Source of Name

This entry was named for Augustus De Morgan.

Sources

• 1951: J.C. Burkill: The Lebesgue Integral ... (previous) ... (next): Chapter $\text {I}$: Sets of Points: $1 \cdot 1$. The algebra of sets
• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 3$: Unions and Intersections of Sets: Exercise $3.6 \ \text{(d)}$