Show that $W$ equivalence relation on $\mathbb{R}^2$

Question

Define relation $W$ on $\mathbb{R}^2$ by $(x_1,y_1)W(x_2,y_2)$ whenever $x_1-y_1=x_2-y_2$.

Show that $W$ is an equivalence relation on $\mathbb{R}^2$.

I believe it is reflexive, not sure about transitive or symmetry.

Also if anybody can tell me about the equivalence classes for this relation, it would also be appreciated.

Thanks.

Answer 1

In general, given a function $f$, we can define a relation $x R_f y$ iff $f(x) = f(y)$. It is easy to see that $x R_f x$, and that if $x R_f y$ then $y R_f x$. And if we have $x R_f y$ and $y R_f z$, then we have $x R_f z$ hence $R_f$ is an equivalence relation.

Furthermore, the equivalence classes are given by $f^{-1}(r)$ for all values $r$ in the range of $f$.

In the example above, we can take $f((x,y)) = x-y$ and note that $W = R_f$.

The equivalence classes are given by $f^{-1}(r) = \{ (x,y) | x-y=r \}$, where $r \in \mathbb{R}$.

(Note: It is not really helpful, but such a function always exists for an equivalence relation, we can define $f(x) = [x]$, the equivalence class of $x$.)