De Morgan's Laws (Set Theory)/Set Complement/Complement of Union/Venn Diagram

Theorem

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

Proof

Demonstration by Venn diagram:

$\overline T_1$ is depicted in yellow and $\overline T_2$ is depicted in red.

Their intersection, $\overline T_1 \cap \overline T_2$, is depicted in orange.

As can be seen by inspection, this also equals the complement of the union of $T_1$ and $T_2$.

Sources

• 2008: David Joyner: Adventures in Group Theory (2nd ed.) ... (previous) ... (next): Chapter $1$: Elementary, my dear Watson: $\S 1.2$: Elements, my dear Watson: Ponderable $1.2.1 \ \text{(d)}$