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