De Morgan's Laws (Set Theory)/Relative Complement

Theorem

Let $S, T_1, T_2$ be sets such that $T_1, T_2$ are both subsets of $S$.

Then, using the notation of the relative complement:

Complement of Intersection

$\relcomp S {T_1 \cap T_2} = \relcomp S {T_1} \cup \relcomp S {T_2}$

Complement of Union

$\relcomp S {T_1 \cup T_2} = \relcomp S {T_1} \cap \relcomp S {T_2}$

General Case

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:

Complement of Intersection

$\ds \relcomp S {\bigcap \mathbb T} = \bigcup_{H \mathop \in \mathbb T} \relcomp S H$

Complement of Union

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

Family of Sets

Let $S$ be a set.

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

Then:

Complement of Intersection

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

Complement of Union

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

Source of Name

This entry was named for Augustus De Morgan.