De Morgan's Laws (Set Theory)/Set Complement/Complement of Intersection/Proof 2

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Theorem

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


Proof

\(\displaystyle \) \(\) \(\displaystyle x \in \overline {T_1 \cap T_2}\)
\(\displaystyle \) \(\leadstoandfrom\) \(\displaystyle x \notin \paren {T_1 \cap T_2}\) Definition of Set Complement
\(\displaystyle \) \(\leadstoandfrom\) \(\displaystyle \neg \paren {x \in T_1 \land x \in T_2}\) Definition of Set Intersection
\(\displaystyle \) \(\leadstoandfrom\) \(\displaystyle x \in \overline {T_1} \lor x \in \overline {T_2}\) Definition of Set Complement
\(\displaystyle \) \(\leadstoandfrom\) \(\displaystyle x \in \overline {T_1} \cup \overline {T_2}\)

By definition of set equality:

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


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