Sum-free set

Remarks

Equivalently, $(A + A) \cap A = \emptyset$, where $A + A$ is the sumset. The set of odd integers is sum-free in $\Z$: if $x$ and $y$ are odd, then $x + y$ is even, so it is not in the set. The set $\set{1, 2, 4}$ is not sum-free: with $x = y = 2$ one gets $x + y = 4$, and repeated elements are allowed. For a positive integer $N$, the upper half $\set{n : N/2 < n \le N}$ of $\set{1, \dots, N}$ is sum-free, since any sum of two of its elements exceeds $N$. The set $\set{0}$ is never sum-free, since $0 + 0 = 0$; so a sum-free subset of a group does not contain $0$.