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

## Theorem

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

Then:

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

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

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

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

## Proof 1

 $\displaystyle \overline {T_1 \cap T_2}$ $=$ $\displaystyle \mathbb U \setminus \paren {T_1 \cap T_2}$ Definition of Set Complement $\displaystyle$ $=$ $\displaystyle \paren {\mathbb U \setminus T_1} \cup \paren {\mathbb U \setminus T_2}$ De Morgan's Laws: Difference with Intersection $\displaystyle$ $=$ $\displaystyle \overline {T_1} \cup \overline {T_2}$ Definition of Set Complement

$\blacksquare$

## Proof 2

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

By definition of set equality:

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

$\blacksquare$

## Proof 3

 $\displaystyle \map \complement {\map \complement A \cup \map \complement B}$ $=$ $\displaystyle \map \complement {\map \complement A} \cap \map \complement {\map \complement B}$ De Morgan's Laws: Complement of Union $\displaystyle$ $=$ $\displaystyle A \cap B$ Complement of Complement $\displaystyle \leadstoandfrom \ \$ $\displaystyle \map \complement {\map \complement {\map \complement A \cup \map \complement B} }$ $=$ $\displaystyle \map \complement {A \cap B}$ taking complements of both sides $\displaystyle \leadstoandfrom \ \$ $\displaystyle \map \complement A \cup \map \complement B$ $=$ $\displaystyle \map \complement {A \cap B}$ Complement of Complement

$\blacksquare$

## 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 $T_2$.

## Source of Name

This entry was named for Augustus De Morgan.