  In this paper we present a procedure which allows to transform a subset $A$
of $\mathbb{Z}_{p}$ into a set $ A'$ such that $ |2\hspace{0.15cm}\widehat{}
A'|\leq|2\hspace{0.15cm}\widehat{} A | $, where $2\hspace{0.15cm}\widehat{} A$
is defined to be the set $\left\{a+b:a\neq b,\;a,b\in A\right\}$. From this
result, we get some lower bounds for $ |2\hspace{0.15cm}\widehat{} A| $.
Finally, we give some remarks related to the problem for which sets $A\subset
\mathbb{Z}_{p}$ we have the equality $|2\hspace{0.15cm}\widehat{} A|=2|A|-1$.
