Prove that if a set of whole numbers is closed for subtraction, it is closed for addition.

Question

The set is a subset of $\mathbb{Z}$. So far, I only know that for $x, y \in A : (x-y) + y \in A$, But I don't know how to prove that $(x + y)$ is also in $A$ based on this? Any direction would be appreciated very much.

Answer 1

Let $A$ be the given set. Pick $x \in A$. Then $0=x-x \in A$. For any $a \in A$ we have $-a=0-a \in A$. Hence $a+b=a-(-b) \in A$ for all $a,b \in A$.