Equivalence classes have the following properties:In $\displaystyle \mathbb{C}[X]$ define a relation by:

$\displaystyle P_1$~$\displaystyle P_2$ if $\displaystyle P_1-P_2$ is divisible by $\displaystyle X-1$.

Show that ~ is an equivalence relation.

1). Reflexivity a~a

2). Symmetry if a~b then b~a

3). Transitivity if a~b and b~c then a~c

1). Reflexivity: $\displaystyle a-a=0=0(X-1)$ so reflexivity holds.

2). Symmetry: if $\displaystyle a-b=m(X-1)$ then $\displaystyle b-a=-m(X-1)$ so symmetry holds.

3). Transitivity: if $\displaystyle a-b=m(X-1)$ and $\displaystyle b-c=q(X-1)$ then $\displaystyle a-b+b-c=(m+q)(X-1)$ so transitivity holds.

I can see that this is going to have something to do with remainders, but i'm not really sure what. Surely remainders can be repeated?Let A denote the set of equivalence classes of ~ in $\displaystyle \mathbb{C}[X]$.

Find an explicit bijection on $\displaystyle A \rightarrow \mathbb{C}$.

Hint: What if the relation were "$\displaystyle P_1$ ~$\displaystyle P_2$ if $\displaystyle P_1-P_2$ is divisible by $\displaystyle X$"?