How would I show that these two sets, involving quotient maps, are equal?

Question

I'm self studying Rotman's Algebraic Topology and I've come across this problem:

I've been looking at this for a bit and I can't figure out how to show that the hint is true.

I see that $$v^{-1}v(C) = \{ u \in X \coprod Y: v(u) \in v(C)\}$$, but beyond this I'm having trouble seeing how the hint was formed.

In the bottom image I've included the graph I've been using to try to make sense of this.

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Answer 1

Suppose $z\in\nu^{-1}(\nu (C))$. This is equivalent to saying that there exists $c\in C$ such that $\nu(z)=\nu(c)$. We have four possibilities.

If $z\in Y$ and $c\in Y$, then $z=c$, and $z\in C\cap Y$.

If $z\in Y$ and $c\in X$, then $f(c)=z$ and $z\in f(C)=f(C\cap A)$.

If $z\in X$ and $c\in Y$, then $f(z)=c$ and $z\in f^{-1}(C)=f^{-1}(C\cap Y)$.

Now, if $z\in X$ and $c\in X$, we have one of the following: either $f(z)=f(c)$ and $z\in f^{-1}(f(C))=f^{-1}(f(C\cap A))$, OR $z=c$ and $z\in C\cap X$.

Since this exhausts all possibilities, we have proved the hint.