Union with Universe

Theorem

The union of a set with the universe is the universe:

$\mathbb U \cup S = \mathbb U$

Proof

\(\ds S\)

\(\subseteq\)

\(\ds \mathbb U\)

Definition of Universe (Set Theory)

\(\ds \leadstoandfrom \ \ \)

\(\ds \mathbb U \cup S\)

\(=\)

\(\ds \mathbb U\)

Union with Superset is Superset‎

$\blacksquare$

Also see

• Intersection with Universe

Sources

• 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{I}$: Sets and Functions: Exercise $\text{B ii}$
• 2008: Paul Halmos and Steven Givant: Introduction to Boolean Algebras ... (previous) ... (next): $\S 2$
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): algebra of sets: $\text {(iv)}$