For $A$ any subset of vector space $X$, is it true that $A+A=2A$?

Question

It seems a trivial question:

Prove/disprove: If we have a vector space $X$, then for any subset $A$ of $X$, we have $A+A =2A$.

It seems that $2A$ is always subset of $A+A$, but I don't think $A+A$ is subset of $2A$.

I am thinking in set of integers modulo $p$, for $p$ a prime, as a counterexample.

Am I right?

Answer 1

Consider $A = \{v, -v\}$ for some vector $v\ne 0$.