Equivalence class finding?

Question

$c_1=a+ib$, $ $ $c_2 = x+iy$
$c_1 R c_2 <=> rec_1 = rec_2 $ $ $
Find all equivalence classes. I have no idea how to write them properly.

Two complex number are in same class if they have the same real part. So an equivalence class of $z=x+iy$ is given by ${z\in \mathbb{C}:\Re(z)=x}$ (this is a vertical line in complex plane) so quotient space isomorphic to $\mathbb{R}$ — user160738, Nov 05 '16 at 08:16

score 1 · Accepted Answer · answered Nov 05 '16 at 08:22

If $R$ is a relation on $X$ characterized by $xRy\iff f(x)=f(y)$ where $f$ is a function with domain $X$, then $R$ is an equivalence relation.

The equivalence class represented by $x\in X$ takes the form:

$$\{y\in X\mid f(x)=f(y)\}$$

Apply that on $X=\mathbb C$ and the function prescribed by $z\mapsto\text{Re}z$.

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