addition on sets

Question

I would like to know if there are known methods to solve arithmetical operation on sets, like $A+X=B$ where $A+X=\{a+x,a \in A, x \in X\}$. I suppose that even if this can't be solved for the general case, there could be some known results with additional hypotheses? (I assume here that all sets are subsets of $\mathbb{R}$, or $\mathbb{Z}$ would be nice too).

Thanks in advance for any hint :)

Could you elaborate on what is actually your question? What do you mean by "solving an arithmetical operation"? — wonko, Jun 21 '17 at 08:26
@wonko I think: for fixed sets $A,B$ find sets $X$ such that $A+X=B$ is true. — drhab, Jun 21 '17 at 08:36

score 1 · Answer 1 · answered Jun 21 '17 at 09:05

Just some thoughts about solving $A+X=B$ where $A,B$ are fixed subsets of a set that is equipped with addition.

You can start with the observation that for every $x\in X$ we need $x+A\subseteq B$.

This invites to start with $X_0:=\{x\mid x+A\subseteq B\}$ as the largest set that satisfies $A+X_0\subseteq B$.

If $Y$ is a solution then it must be a subset of $X_0$ so that in that case $B=A+Y\subseteq A+X_0\subseteq B$. Then consequently also $X_0$ is a solution.

So if $A+X_0\neq B$ then there are no solutions.

Thanks for your reply. Well, it could be somewhat complicated to compute the largest set $X_0$ of elements such that $A+X_0 \subset B$, but I assume that once this is done (?), the answer will be known. — user456738, Jun 21 '17 at 11:49

addition on sets

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