Intersection with Universal Set

Theorem

The intersection of a set with the universal set is the set itself:

$\mathbb U \cap S = S$

Proof

\(\ds S\)

\(\subseteq\)

\(\ds \mathbb U\)

Definition of Universal Set

\(\ds \leadstoandfrom \ \ \)

\(\ds \mathbb U \cap S\)

\(=\)

\(\ds S\)

Intersection with Subset is Subset‎

$\blacksquare$

Also see

• Union with Universal Set

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 3$: Unions and Intersections of Sets: Theorem $3.1$
• 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: $\text {(iv)}$
• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): algebra of sets: $\text {(iv)}$