The book I am reading (Szekeres's A Course in Modern Mathematical Physics) makes the following claim:
More generally, any map $φ : X → Y$ defines an equivalence relation $R$ on $X$ by $aRb$ iff $φ(a) = φ(b)$. The equivalence classes defined by $R$ are precisely the inverse images of the singleton subsets of Y, $$X/R = \{φ^{−1}({y}) | y ∈ Y \},$$ and the map $ψ : Y → X/R$ defined by $ψ(y) = φ^{−1}(\{y\})$ is one-to-one, for if $ψ(y) = ψ(y′)$ then y = y′ – pick any element $x ∈ ψ(y) = ψ(y′)$ and we must have $φ(x) = y = y′$.
I think the equating of $X/R$ to $\{φ^{−1}({y}) | y ∈ Y \}$ is a bit sloppy here. In particular, unless $\varphi$ is onto, we will have $\emptyset \in \{φ^{−1}({y}) | y ∈ Y \}$, and yet $\emptyset \notin X/R$. Of course the rest of the definition would need to be modified too since $\psi$ does not make sense as it stands (without $\varphi$ onto). Am I missing something here?