Equivalent Conditions for Cover by Collection of Subsets

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Theorem

Let $X$ be a set.

Then the following conditions are equivalent for a subset $\CC \subseteq \powerset X$ of the power set of $X$:

$(1): \quad \CC$ is a cover for $X$.

$(2): \quad \ds X = \bigcup \CC$.

$(3): \quad \ds \exists \SS \subseteq \CC: X = \bigcup \SS$.

Proof

$(1)$ implies $(2)$

By definition, $\CC$ covers $X$ if and only if $X \subseteq \ds \bigcup \CC$.

By Union of Subsets is Subset, we have that:

$\ds \bigcup \CC \subseteq X$

since $\CC \subseteq \powerset X$.

By definition of set equality, it follows that $X = \ds \bigcup \CC$.

$\Box$

$(2)$ implies $(3)$

Since $\CC \subseteq \CC$, we can take $\SS = \CC$.

Hence $(3)$ is immediate from $(2)$.

$\Box$

$(3)$ implies $(1)$

By assumption, $X = \ds \bigcup \SS$ for some $\SS \subseteq \CC$.

By Union is Increasing, we have that:

$\ds \bigcup \SS \subseteq \bigcup \CC$

Hence, $X \subseteq \ds \bigcup \CC$, that is $\CC$ is a cover for $X$.

$\blacksquare$