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

## 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:

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

## Proof 1

Let $T_1, T_2 \subseteq S$.

Then from Union is Smallest Superset:

$T_1 \cup T_2 \subseteq S$

Hence:

 $\ds \relcomp S {T_1 \cup T_2}$ $=$ $\ds S \setminus \paren {T_1 \cup T_2}$ Definition of Relative Complement $\ds$ $=$ $\ds \paren {S \setminus T_1} \cap \paren {S \setminus T_2}$ De Morgan's Laws: Difference with Union $\ds$ $=$ $\ds \relcomp S {T_1} \cap \relcomp S {T_2}$ Definition of Relative Complement

$\blacksquare$

## Proof 2

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.