# Set Dualities

# Why

How does taking complements relate to forming
unions and intersections.

## Complements of unions or intersections

Let $E$ denote a set.
Let $A$ and $B$ denote sets and $A, B \subset
E$.
All complements are taken with respect to $E$.
The following are known as
DeMorgan’s Laws.

$\complement{A \cup B} = \complement{A} \cap
\complement{B}$

$\complement{A \cap B} = \complement{A} \cup
\complement{B}$

## Principle of duality

As a result of DeMorgan’s Laws
and basic facts about complements (see Set Complements) theorems about sets often come in pairs.
In other words, given an inclusion or identity
relation involving complements, unions and
intersections of some set (above $E$) if we
replace all sets by their complemnets, swap
unions and intersections, and flip all inclusions
we obtain another, true, result.
The correspondence is called the
principle of duality for
sets.