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

Theorem

Let $T_1, T_2$ be subsets of a universe $\mathbb U$.

Let $\overline T_1$ denote the set complement of $T_1$.

Then:

$\overline {T_1 \cap T_2} = \overline T_1 \cup \overline T_2$

$\overline {T_1 \cup T_2} = \overline T_1 \cap \overline T_2$

Let $\mathbb T$ be a set of sets, all of which are subsets of a universe $\mathbb U$.

Then:

$\ds \map \complement {\bigcap \mathbb T} = \bigcup_{H \mathop \in \mathbb T} \map \complement H$

$\ds \map \complement {\bigcup \mathbb T} = \bigcap_{H \mathop \in \mathbb T} \map \complement H$

Let $\family {S_i}_{i \mathop \in I}$ be a family of sets, all of which are subsets of a universe $\Bbb U$.

Then:

$\ds \map \complement {\bigcap_{i \mathop \in I} S_i} = \bigcup_{i \mathop \in I} \map \complement {S_i}$

$\ds \map \complement {\bigcup_{i \mathop \in I} S_i} = \bigcap_{i \mathop \in I} \map \complement {S_i}$

This entry was named for Augustus De Morgan.