The question is to determine whether $\displaystyle \mathcal{A}$ is a partition of the set $\displaystyle A$.

$\displaystyle A = \mathbb{R}$ and $\displaystyle \mathcal{A} = \{S_y : y \epsilon \mathbb{R}\}$, where $\displaystyle S_y = \{x \epsilon \mathbb{R} : |x| = |y|\}$.

I understand what it means to be a partition, but I'm not sure what $\displaystyle \mathcal{A}$, or particularly, $\displaystyle S_y$ means.

Does this mean $\displaystyle \mathcal{A} = \{ \{1, 1\}, \{1.1, 1.1\}, \{1.2, 1.2\}, ..., \{-1, 1\}, \{-1.1, 1.1\}, \{-1.2, 1.2\}, ... \{1, -1\}, \{1.1, -1.1\}, ... \{-1, -1\}, ... \{-5.4, 5.4\} \}$? Where I just find any real number pair so that their absolute values are equal?