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


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: $\text {(iv)}$
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