abstract algebra - 1-1 Correspondence $(S times T) times U$ and $S times (T times U)$

(Herstein section 1.2 problem 3)

If $S, T, U$ are nonempty sets, prove that there exists a one-to-one correspondence $(S \times T) \times U$ and $S \times (T \times U)$.

An element of $(S \times T) \times U$ is of the form $((s,t),u)$ and for $S \times (T \times U)$ an element is of the form $(s,(t,u))$.

I am unsure of such a function. The only thing that comes to mind is that given an element of the form $((s,t),u)$ take the $T$ value from $S \times T$ and then take the $U$ value from $(S \times T) \times U$ to obtain an element of $T \times U$ and then take that value and cross it with the value from $S$ to get an element from $S \times (T \times U)$.

But I am highly unsure of this function as it is literally looking at the form of the elements and essentially "swapping the parentheses".

Answer

The function $f:(S\times T)\times U\to S\times(T\times U)$ prescribed by

$$\langle\langle s,t\rangle,u\rangle\mapsto\langle s,\langle t,u\rangle\rangle$$ is evidently surjective and can also be proved to be injective (can you do that?).

So $f$ is a bijection whence represents a one-to-one correspondence between domain and codomain of $f$.