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

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## Theorem

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

## Proof

\(\ds \map \complement {T_1 \cup T_2}\) | \(=\) | \(\ds \mathbb U \setminus \paren {T_1 \cup T_2}\) | Definition of Set Complement | |||||||||||

\(\ds \) | \(=\) | \(\ds \paren {\mathbb U \setminus T_1} \cap \paren {\mathbb U \setminus T_2}\) | De Morgan's Laws: Difference with Union | |||||||||||

\(\ds \) | \(=\) | \(\ds \map \complement {T_1} \cap \map \complement {T_2}\) | Definition of Set Complement |

$\blacksquare$

## Sources

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- 2008: Paul Halmos and Steven Givant:
*Introduction to Boolean Algebras*... (previous) ... (next): $\S 2$