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The union of two sets $\mathcal{A}$ and $\mathcal{B}$ is the set of elements
that are in at least one of the two sets,  and is designated as
$\mathcal{A\cup B}$. This operation is commutative
$\mathcal{A\cup B = B\cup A}$ and associative $\mathcal{(A\cup B)\cup C =
A\cup(B\cup C)}$.  If $\mathcal{A\subseteq B}$, then
$\mathcal{A\cup B = B}$. It then follows that $\mathcal{A\cup A = A}$,
$\mathcal{A\cup\{\emptyset\} = A}$ and $\mathcal{J\cup A = J}$. 

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