 Difference set

For finite sets $A,B\subseteq G$ in an abelian group $G$, their difference set isHere $-b$ is the additive inverse in $G$. The difference set records possible displacements from a member of $B$ to a member of $A$; it is distinct from the set difference $A\setminus B$.

For the integer sets $A=\{0,1,2,3\}$ and $B=\{0,1,4,10\}$,The positive differences for $B$ are $1,3,4,6,9,10$, each obtained once; reversing each ordered pair supplies the corresponding negative difference. Every nonempty finite set $S$ has $0\in S-S$, with $|S|$ representations $(s,s)$. Swapping the members of an ordered pair also shows $S-S=-(S-S)$. In the cyclic group $\mathbb{Z}/6\mathbb{Z}$, the subgroup $H=\{0,2,4\}$ satisfies $H-H=H$ as well as $H+H=H$.