Complement of Empty Set is Universal Set

Theorem

The complement of the empty set is the universal set:

$\map \complement \O = \mathbb U$


Proof

Substitute $\mathbb U$ for $S$ in $\relcomp S \O = S$ from Relative Complement of Empty Set.

$\blacksquare$


Also see


Sources

  • 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{I}$: Sets and Functions: Exercise $\text{B i}$
  • 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 6.6$: Subsets
  • 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: $\text {(viii)}$
  • 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): complement, complementation
  • 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): algebra of sets: $\text {(viii)}$
  • 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): complement, complementation
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