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

## 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

$\blacksquare$