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

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## Theorem

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

## Proof

\(\displaystyle \overline {T_1 \cap T_2}\) | \(=\) | \(\displaystyle \mathbb U \setminus \paren {T_1 \cap T_2}\) | Definition of Set Complement | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \paren {\mathbb U \setminus T_1} \cup \paren {\mathbb U \setminus T_2}\) | De Morgan's Laws: Difference with Intersection | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \overline {T_1} \cup \overline {T_2}\) | Definition of Set Complement |

$\blacksquare$

## Sources

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