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

$\blacksquare$

Also see

• Intersection with Complement is Empty iff Subset

Sources

• 1955: John L. Kelley: General Topology ... (previous) ... (next): Chapter $0$: Subsets and Complements; Union and Intersection: Theorem $1$
• 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{I}$: Sets and Functions: Exercise $\text{B vi}$