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

## 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

$\blacksquare$