Sumset

Remarks

In $\Z$, take $A = \set{0, 1, 2}$ and $B = \set{0, 5}$. Then $A + B = \set{0, 1, 2, 5, 6, 7}$, since each sum $a + b$ lies in this set and each element of it is such a sum. The set $\set{a + a : a \in A} = \set{0, 2, 4}$ is not the sumset $A + A$, because it misses sums of two different elements such as $0 + 1$. The iterated sumset $2A = A + A$ is different from the dilate $2 \cdot A = \set{2a : a \in A}$. For finite nonempty $A, B \subseteq \Z$ one has $\abs{A + B} \ge \abs{A} + \abs{B} - 1$, with equality for arithmetic progressions of the same common difference. For subsets of $\R^n$ the sumset is called the Minkowski sum.