Relative Complement of Empty Set

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Theorem

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

$\complement_S \left({\varnothing}\right) = S$


Proof

\(\displaystyle \complement_S \left({\varnothing}\right)\) \(=\) \(\displaystyle S \setminus \varnothing\) $\quad$ Definition of Relative Complement $\quad$
\(\displaystyle \) \(=\) \(\displaystyle S\) $\quad$ Set Difference with Empty Set is Self $\quad$

$\blacksquare$


Also see


Sources