how to show the equivalence relation

Question

Consider the relation ∼ on $\mathbb{R}\times \mathbb{R}$ defined as follows for all $(x, y),(a, b) ∈\mathbb{R}\times \mathbb{R}$. $(x, y) \sim (a, b)$ if and only if $x − a = y − b$. Show that $∼$ is an equivalence relation and describe geometrically how this relation partitions the plane.

Answer 1

The reflexivity is straightforward. For the symmetry, if $x-a=y-b$ then $y-b=x-a$ and so $(x,y)\sim (a,b)$.

Ad regards the transitivity, $x-a=y-b$ and $a-\alpha=b-\beta$ then $x-\alpha=(x-a)+(a-\alpha)=(y-b)+(b-\beta=y-\beta$ and so $(x,y)\sim (\alpha,\beta)$

You can observe that an equivalence class is the following:

$[(x_0,y_0)]=\{(x_0+k, y_0+k): k\in \mathbb{R}\}$

$=\{(x, x+(y_0-x_0)) : x\in \mathbb{R}\}$

that is the graph of a line. This line through the point $(x_0,y_0)$ and it is parallel to the bisect of the first and four quadrant. Thus the quotient set is the set of this lines