Sometimes called the ‘big union,’ the sumset of a set \(A\) is the set of the elements of the elements of \(A\), i.e., the union of its elements. In set-theoretic notation this is

\[\bigcup A = \left\{x \,\left| \,\exists y (x \in y \mbox{ and } y \in A \right.\right\}\]

It is an axiom of formal set theory that for any set \(A\) the sumset of \(A\) exists.