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

• Relative Complement with Self is Empty Set

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