
Partition of a set
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?

Re: Partition of a set
$\displaystyle S_x = \{ x, x \} \text{ for } x \in \mathbb{R}, x \ne 0. \ S_0 = \{ 0 \}. \ \mathcal{A} = \{ S_y : y \in \mathbb{R} \}.$
$\displaystyle \text{Since } S_x = S_{x} \text{ for } x \ne 0, \text{ have that }$
$\displaystyle \mathcal{A} = \{ S_y : y \in \mathbb{R} \} = \{ S_y : y \in \mathbb{R}, y \ge 0 \}.$
Are those sets in $\displaystyle \mathcal{A}$ disjoint? Nonempty? Does their union equal the whole set (in this case, $\displaystyle \mathbb{R}$)? If so, then $\displaystyle \mathcal{A}$ is a partition of $\displaystyle \mathbb{R}.$

The only potentially confusing thing here is that the original definition of $\displaystyle \mathcal{A}$ had redundancies  two different names for the same set  and so that might seem to make it not a partition. However, a set with redundant (even if differently named) elements is the same as the set with unique elements.
i.e. If a = 2, b = 2, then U = {a, b} = { 2, 2 } = { 2 } = { a }.
Likewise: If A = {5, 2}, B = {5, 2}, then V = {A, B} = { {5,2}, {5,2} } = { {5,2} } = { A }.