De Morgan's Laws (Set Theory)

This proof is about De Morgan's Laws in the context of Set Theory. For other uses, see De Morgan's Laws.

Theorem

De Morgan's laws are a collection of results in set theory as follows.

Set Difference

$S \setminus \paren {T_1 \cap T_2} = \paren {S \setminus T_1} \cup \paren {S \setminus T_2}$
$S \setminus \paren {T_1 \cup T_2} = \paren {S \setminus T_1} \cap \paren {S \setminus T_2}$

Relative Complement

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

Set Complement

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

Also known as

De Morgan's Laws are also known as the De Morgan formulas.

Some sources, whose context is that of logic, refer to them as the laws of negation.

Some sources refer to them as the duality principle.

Source of Name

This entry was named for Augustus De Morgan.

Historical Note

Augustus De Morgan proposed what are now known as De Morgan's laws in $1847$, in the context of logic.

They were subsequently applied to the union and intersection of sets, and in the context of set theory the name De Morgan's laws has been carried over.