Relative Complement of Empty Set

Theorem

The relative complement of the empty set is the set itself:

$\relcomp S \O = S$


Proof

\(\ds \relcomp S \O\) \(=\) \(\ds S \setminus \O\) Definition of Relative Complement
\(\ds \) \(=\) \(\ds S\) Set Difference with Empty Set is Self

$\blacksquare$


Also see


Sources

  • 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 5$: Complements and Powers
  • 1965: A.M. Arthurs: Probability Theory ... (previous) ... (next): Chapter $1$: Set Theory: $1.3$: Set operations
  • 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 3$: Unions and Intersections of Sets: Theorem $3.2$
  • 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 1$. Sets; inclusion; intersection; union; complementation; number systems
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