Sumset

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In additive number theory, the sumset of two sets of natural numbers A and B is defined as the set of all sums of an element from A with an element from B, together with the elements of A and of B. That is,

$A \oplus B = \{a, b, a+b : (a \in A, b \in B)\}$

$A \oplus B = (A + B) \cup A \cup B.$

If $0 \in A \cap B$ , then $A \oplus B$ coincides with $A + B$ , that is, the sets of sums alone. The notation is not that of the direct sum in abstract algebra.