De Morgan's Laws (Set Theory)/Set Complement/Complement of Intersection/Proof 3
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Theorem
- $\overline {T_1 \cap T_2} = \overline T_1 \cup \overline T_2$
Proof
| \(\ds \map \complement {\map \complement A \cup \map \complement B}\) | \(=\) | \(\ds \map \complement {\map \complement A} \cap \map \complement {\map \complement B}\) | De Morgan's Laws: Complement of Union | |||||||||||
| \(\ds \) | \(=\) | \(\ds A \cap B\) | Complement of Complement | |||||||||||
| \(\ds \leadstoandfrom \ \ \) | \(\ds \map \complement {\map \complement {\map \complement A \cup \map \complement B} }\) | \(=\) | \(\ds \map \complement {A \cap B}\) | taking complements of both sides | ||||||||||
| \(\ds \leadstoandfrom \ \ \) | \(\ds \map \complement A \cup \map \complement B\) | \(=\) | \(\ds \map \complement {A \cap B}\) | Complement of Complement |
$\blacksquare$
Sources
- 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $1$: Theory of Sets: $\S 3$: Set Operations: Union, Intersection and Complement: Theorem $3.2$