Definition:Cover of Set

Definition 1
Let $S$ be a set.

A cover for $S$ is a set of sets $\mathcal C$ such that:
 * $\displaystyle S \subseteq \bigcup \mathcal C$

where $\bigcup \mathcal C$ denotes the union of $\mathcal C$.

We say that $S$ is covered by $\mathcal C$.

Definition 2
Let $\mathcal S$ be a set of subsets of a set $S$.

Then $S$ is covered by $\mathcal S$ iff:


 * $\{\complement (X): X \in \mathcal S\}$

is free, where $\complement (X)$ denotes the complement of $X$ in $S$.

Equivalence of Definitions
That the above definitions are equivalent is shown on Equivalence of Cover Definitions.

Also known as
A cover is also known as a covering.