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

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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

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

$\blacksquare$


Proof 2

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

By definition of set equality:

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

$\blacksquare$


Proof 3

\(\ds \map \complement {\map \complement A \cup \map \complement B}\) \(=\) \(\ds \map \complement {\map \complement A} \cap \map \complement {\map \complement B}\) 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

$\blacksquare$


Demonstration by Venn Diagram

DeMorganComplementIntersection.png

$\overline T_1$ 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.


Sources


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