Relations exercise Charles C. Pinter

Question

let $ f: A \to B $ a surjective function. Suppose that $R$ is a relation of equivalence in $A$ such that $ R_f \subset R $ and that $S$ is a relation of equivalence in $B$

Definition:

$\overrightarrow {f} (R)= \{ (f(x), f(y) ): (x,y) \in R \}$

Prove:

a)$ \overrightarrow {f} (R) $ is a equivalence of relation in $B$

$$\begin{array}{crl} \text{let } x \in \overrightarrow {f} (R) &\Rightarrow & (x,x) \in R\\ &\Rightarrow & (x,x) \in R_f\\ &\Rightarrow&\ f(x)=f(y)=b\\ &\Rightarrow&\ (x,x) \in B \times B\\ & \end{array}$$

Can you help me check this proof?

Answer 1

This proof makes little sense to me. In your third line you say $f(x) = f(y) = b.$ What is $y$? What is $b$?