Canonical injection is compatible with induced equivalence relation?

Question

Let $R$ be an equivalence relation on a set $E$, let $A$ be a subset of $E$, let $j : A \to E$ be the canonical injection, and let $R_A$ be the equivalence relation induced by $R$ on $A$ (that is, the inverse image of $R$ under $j$). According to Bourbaki:

The injection $j$ is obviously compatible with the relations $R_A$ and $R$. (Theory of Sets, p. 119)

Why?

Let $g$ be the canonical projection of $A$ onto $R/A$. If the statement above is true, and if $g(x) = g(x^\prime)$, then $j(x) = j(x^\prime)$. That is, $x = x^\prime$. Therefore $g(x) = \{x\}$. But that isn't true in general.

What is the error?

Answer 1

There doesn't seem to be anything in your assumptions that requires $j(x)=j(x')$. What compatibility requires is just that

If $x\mathrel{R_A} x'$ then $j(x)\mathrel R j(x')$

which is obviously true. Or, equivalently,

If $g(x)=g(x')$ then $g(j(x))=g(j(x'))$

which is pretty much vacuous because $j$ is by definition the identity on $A$.