Relative Complement with Self is Empty Set

Theorem

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

$\relcomp S S = \O$


Proof

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

$\blacksquare$


Also see


Sources

  • 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 5$: Complements and Powers
  • 1961: John G. Hocking and Gail S. Young: Topology ... (previous) ... (next): A Note on Set-Theoretic Concepts
  • 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 $3.2$
  • 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 2$: Sets and functions: Sets
  • 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 1$. Sets; inclusion; intersection; union; complementation; number systems
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