I don't get what is the function so $x = +$ and $- y$ so there's two $y$? or does it mean $x = x + y$ and $x = x - y$. I don't know how to read this and what would its relation look like? $\{...(1,-1),(2,-2)...\}$etc? and cause of that I don't understand the explanation of why its symmetric and why its not anti-symmetric.
On
This is an attempt to define a relation, that is, we are defining how two real numbers relate to each other via a definition (somewhat similar to how we define order relationship in the real numbers).
$$x \equiv_R y \iff x=y \text{ or } x=-y$$
For example, $1 \equiv_R -1$ and $1 \equiv_R 1$
Can you see why it does not satisfy anti-symmetry?
No operation involved here.
One defines the relation $R$ in the following way:
$$x \equiv_R y \iff x = \pm y$$
This means: either $x = y$ or $x = -y$
For example:
$$3 \equiv_R 3$$ $$3 \equiv_R -3$$
In fact, we can write this relation as:
$$R = \{(a,a) \in \mathbb{R^2}\} \cup \{(a,-a) \in \mathbb{R^2}\} $$