Sumset

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 of A with an element of B, together with the elements of A and of B. That is,

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

or

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

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.

See also: Schnirelmann density.