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

• Relative Complement of Empty Set

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