Equivalence of Definitions of Cover of Set

Theorem

Let $S$ be a set.

The following definitions of the concept of Cover of Set are equivalent:

Definition 1

A cover for $S$ is a set of sets $\CC$ such that:

$\ds S \subseteq \bigcup \CC$

where $\bigcup \CC$ denotes the union of $\CC$.

Definition 2

A cover for $S$ is a set of sets $\CC$ such that:

$\forall s \in S : \exists C \in \CC : x \in C$

Proof

Definition 1 Implies Definition 2

Let $\CC$ be a set of sets such that:

$\ds S \subseteq \bigcup \CC$

where $\bigcup \CC$ denotes the union of $\CC$.

By definition of subset:

$\forall s \in S : s \in \ds \bigcup \CC$

By definition of set union:

$\forall s \in S : \exists C \in \CC : s \in C$

$\Box$

Definition 2 Implies Definition 1

Let $\CC$ be a set of sets such that:

$\forall s \in S : \exists C \in \CC : s \in C$

From Set is Subset of Union (General Result):

$\forall C \in \CC : C \subseteq \ds \bigcup \CC$

By definition of subset:

$\forall s \in S : s \in \ds \bigcup \CC$

By definition of subset:

$S \subseteq \ds \bigcup \CC$

$\blacksquare$