# De Morgan's Laws (Set Theory)/Relative Complement/Complement of Union/Proof 2

## Theorem

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

## Proof

Let $x \in S$ througout.

 $\ds$  $\ds x \in \relcomp S {T_1 \cup T_2}$ $\ds$ $\leadsto$ $\ds x \notin \paren {T_1 \cup T_2}$ Definition of Relative Complement $\ds$ $\leadsto$ $\ds \neg \paren {x \in T_1 \lor x \in T_2}$ Definition of Set Union $\ds$ $\leadsto$ $\ds x \notin T_1 \land x \notin T_2$ De Morgan's Laws: Conjunction of Negations $\ds$ $\leadsto$ $\ds x \in \relcomp S {T_1} \land x \in \relcomp S {T_2}$ Definition of Relative Complement $\ds$ $\leadsto$ $\ds x \in \relcomp S {T_1} \cap \relcomp S {T_2}$ Definition of Set Intersection $\ds$ $\leadsto$ $\ds \relcomp S {T_1 \cup T_2} \subseteq \relcomp S {T_1} \cap \relcomp S {T_2}$ Definition of Subset

 $\ds$  $\ds x \in \relcomp S {T_1} \cap \relcomp S {T_2}$ $\ds$ $\leadsto$ $\ds x \in \relcomp S {T_1} \land x \in \relcomp S {T_2}$ Definition of Set Intersection $\ds$ $\leadsto$ $\ds x \notin T_1 \land x \notin T_2$ Definition of Relative Complement $\ds$ $\leadsto$ $\ds \neg \paren {x \in T_1 \lor x \in T_2}$ De Morgan's Laws: Conjunction of Negations $\ds$ $\leadsto$ $\ds x \notin \paren {T_1 \cup T_2}$ Definition of Set Union $\ds$ $\leadsto$ $\ds x \in \relcomp S {T_1 \cup T_2}$ Definition of Relative Complement $\ds$ $\leadsto$ $\ds \relcomp S {T_1} \cap \relcomp S {T_2} \subseteq \relcomp S {T_1 \cup T_2}$ Definition of Set Intersection

By definition of set equality:

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

$\blacksquare$

## Source of Name

This entry was named for Augustus De Morgan.