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

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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

Let $T_1, T_2 \subseteq S$.

Then from Union is Smallest Superset:

$T_1 \cup T_2 \subseteq S$

Hence:

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

$\blacksquare$


Source of Name

This entry was named for Augustus De Morgan.


Sources