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

Theorem

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

Proof

Demonstration by Venn diagram:

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

Their intersection, where they overlap, is depicted in orange.

Their union $\overline T_1 \cup \overline T_2$ is the total shaded area: yellow, red and orange.

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

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 3$: Unions and Intersections of Sets: Exercise $3.2$
• 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 $1.2.1 \ \text{(c)}$