De Morgan's Laws (Set Theory)/Relative Complement/Complement of Union/Proof 1

Theorem

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

Proof

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

$\blacksquare$

Source of Name

This entry was named for Augustus De Morgan.