# Complement Union with Superset is Universe

## Theorem

$S \subseteq T \iff \map \complement S \cup T = \mathbb U$

where:

$S \subseteq T$ denotes that $S$ is a subset of $T$
$S \cup T$ denotes the union of $S$ and $T$
$\complement$ denotes set complement
$\mathbb U$ denotes the universal set.

### Corollary

$S \cup T = \mathbb U \iff \map \complement S \subseteq T$

## Proof

 $\ds S$ $\subseteq$ $\ds T$ $\ds \leadstoandfrom \ \$ $\ds S \cap \map \complement T$ $=$ $\ds \O$ Intersection with Complement is Empty iff Subset $\ds \leadstoandfrom \ \$ $\ds \map \complement {S \cap \map \complement T}$ $=$ $\ds \mathbb U$ Complement of Empty Set is Universe $\ds \leadstoandfrom \ \$ $\ds \map \complement S \cup \map \complement {\map \complement T}$ $=$ $\ds \mathbb U$ De Morgan's Laws: Complement of Intersection $\ds \leadstoandfrom \ \$ $\ds \map \complement S \cup T$ $=$ $\ds \mathbb U$ Complement of Complement

$\blacksquare$