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

## Theorem

Let $T_1, T_2$ be subsets of a universe $\mathbb U$.

Then:

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

where $\overline T_1$ is the set complement of $T_1$.

It is arguable that this notation may be easier to follow:

$\map \complement {T_1 \cup T_2} = \map \complement {T_1} \cap \map \complement {T_2}$

### Corollary

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

## Proof 1

 $\ds \map \complement {T_1 \cup T_2}$ $=$ $\ds \mathbb U \setminus \paren {T_1 \cup T_2}$ 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

$\blacksquare$

## Proof 2

 $\ds$  $\ds x \in \overline {T_1 \cup T_2}$ $\ds$ $\leadstoandfrom$ $\ds x \notin \paren {T_1 \cup T_2}$ Definition of Set Complement $\ds$ $\leadstoandfrom$ $\ds \neg \paren {x \in T_1 \lor x \in T_2}$ Definition of Set Union $\ds$ $\leadstoandfrom$ $\ds \neg \paren {x \in T_1} \land \neg \paren {x \in T_2}$ De Morgan's Laws (Logic): Conjunction of Negations $\ds$ $\leadstoandfrom$ $\ds x \in \overline {T_1} \land x \in \overline {T_2}$ Definition of Set Complement $\ds$ $\leadstoandfrom$ $\ds x \in \overline {T_1} \cap \overline {T_2}$

By definition of set equality:

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

$\blacksquare$

## 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 $T_2$.

## Source of Name

This entry was named for Augustus De Morgan.