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.



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 $x$.)


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