Loomis and Sternberg Problem 1.15
Take an element $v\in A\cap B$ then $v=\alpha+u=\beta+w$ for some $u,w\in M$. Then $\alpha-\beta=w-u\in M$. This means that $\alpha\in \beta+M$. So $A=\alpha+M=(\beta+M)+M=\beta+M=B$. This is implies the result.
Loomis and Sternberg Problem 1.15
Take an element $v\in A\cap B$ then $v=\alpha+u=\beta+w$ for some $u,w\in M$. Then $\alpha-\beta=w-u\in M$. This means that $\alpha\in \beta+M$. So $A=\alpha+M=(\beta+M)+M=\beta+M=B$. This is implies the result.