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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