Definition:Set Complement

Definition
The set complement (or, when the context is established, just complement) of a set $$S$$ in a universe $$\mathbb{U}$$ is defined as:


 * $$\complement \left ({S}\right) = \complement_{\mathbb{U}} \left ({S}\right) = \mathbb{U} \setminus S$$

See the definition of Relative Complement for the definition of $$\complement_{\mathbb{U}} \left ({S}\right)$$.

Thus the complement of a set $$S$$ is the relative complement of $$S$$ in the universe, or the complement of $$S$$ relative to the universe.

A common alternative to the symbology $$\complement \left ({S}\right)$$, which we will sometimes use, is $$\overline S$$.

Notation
No standard symbol for this concept has evolved. There are alternative symbols for $$\complement \left ({S}\right)$$ and $$\overline S$$. $$\mathcal{C} \left ({S}\right)$$ is sometimes encountered, and appears on this website. Another common one is $$S'$$, but it can be argued that the symbol $$'$$ is already overused.

Some authors use $$S^c$$, but that can also been confused with notation used for the conjugate. Another is $$\complement \left({S}\right)$$. Some authors use $$C S$$. Another one is $$S^*$$, and another is $$\tilde S$$. You may encounter others.