Let $ \mathbb Z[x] $ denote the set of all polynomials of the variable $ x $ with integer coefficients, and let $ r $ be such a relation in the set $ \mathbb Z[x] $ that $ \left\langle f, g\right\rangle \in r $ holds if and only if the difference $ f - g $ has all even coefficients.
(a) Show that $ r $ is an equivalence relation.
(b) Identify three different classes of abstraction.
(c) What is the power of the set of all abstraction classes?
(d) Identify all the cardinal numbers that are powers of the abstract classes of this relation.
I think I can solve this problem for (a):
We know that $x_i - y_i$ and $y_i - z_i$ are even; it follows that their sum, i.e., $x_i - z_i$, is also even. So $r$ is transitive. The proof of symmetry is that $g - f = - (f - g)$. And for reflexivity: let's take any polynomial $f \in \mathbb Z[x]$. Then $f - f = 0$, and $0$ is an even number, so $f - f$ has all even coefficients. Therefore $\left\langle f, f\right\rangle \in r$ and the relation $r$ is reflexive. Then $r$ is an equivalence relation.
But I don't know what how to solve next problems. Any help would be much appreciated.