Union of Empty Set

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Theorem

Consider the set of sets $\mathbb S$ such that $\mathbb S$ is the empty set $\varnothing$.


Then the union of $\mathbb S$ is $\varnothing$:

$\displaystyle \mathbb S = \varnothing \implies \bigcup \mathbb S = \varnothing$


Proof

Let $\mathbb S = \varnothing$.

Then from the definition:

$\displaystyle \bigcup \mathbb S = \left\{{x: \exists X \in \mathbb S: x \in X}\right\}$

from which it follows directly:

$\displaystyle \bigcup \mathbb S = \varnothing$

as there are no sets in $\mathbb S$.

$\blacksquare$


Sources