Union of Singleton

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Consider the set of sets $A$ such that $A$ consists of just one set $x$:

$A = \set x$

Then the union of $A$ is $x$:

$\bigcup A = x$


Let $A = \set x$.

From the definition of set union:

$\bigcup \set x = \set {y: \exists z \in \set x: y \in z}$

from which it follows directly that:

$\bigcup \set x = \set {y: y \in x}$

as $x$ is the only set in $\set x$.

That is:

$\bigcup A = x$


Also see