Baire Category and Residual Sets

I posted this problem some time ago, and it didn't elicit any feedback. So, in compliance with MHF Rules, I am now reposing the problem with some additional info. The result is bound to be long-winded; apologies.

Recall the topological notion of a nowhere dense set. There are a number of equivalent definitions of this, but I shall be using the following: "A set $\displaystyle S$ in $\displaystyle \mathbb{R}$ is said to be *nowhere dense *provided that its complement $\displaystyle S^\text{C}$ contains a dense, open set." If a set in $\displaystyle \mathbb{R}$ can be expressed as a union of countably many nowhere dense sets, it is said to be of *first (Baire) category, *and if it cannot be so expressed, it is said to be of *second (Baire) category.* The complement of a set of first category is often called *residual.* It is easy to show that, in $\displaystyle \mathbb{R},$ any residual set is of second category.

I need to prove the following:

Let $\displaystyle A$ and $\displaystyle B$ be any two residual sets in $\displaystyle \mathbb{R}.$ For fixed real numbers $\displaystyle x$ and $\displaystyle y$, the sets $\displaystyle A_x=\{x-a : a\in A\}$ and $\displaystyle B_y = \{y/b : b\in B, b\neq 0\}$ are both residual in $\displaystyle \mathbb{R}$; that is, they are both complements of sets of first category.

I can see how this is true intuitively, but I have never been able to find a rigorous proof. Any guidance anyone can provide would be of great help.