Homeomorphism torus

Question

Let $X=[0,1]\times [0,1] \subset \mathbb{R}^2$ and consider the following partition of $X$:

1. the set $\{(0,0),(1,0),(0,1),(1,1)\}$
2. sets consisting of pairs of points $(x,0)$, $(x,1)$, where $0 < x < 1$;
3. sets consisting of pairs of points $(0,y)$, $(1,y)$, where $0 < y < 1$;
4. sets consisting of a single point $(x,y)$ with $0<x<1$ and $0<y<1$.

This induces a equivalence relation $$(x,y) \sim (w,z) \Leftrightarrow (x,y), (w,z) \hbox{ belong to the same type of set of the partition above}.$$

Let $f=[0,1]\times [0,1] \rightarrow S^1\times S^1$ given by \begin{equation*} f(x,y)=(e^{2\pi i x}, e^{2 \pi i y}). \end{equation*}

I already proved that $(x,y) \sim (w,z)$ then $f(x,y)=f(w,z)$.

My question:

Is it true that $f(x,y)=f(w,z)$ then $(x,y) \sim (w,z)$?

If so, how can we prove this?