Prove that $X \times Y \times Z = X \times (Y \times Z) = (X \times Y) \times Z$.

Question

$X$ and $Y$ are sets.
$X \times Y$ denotes the cartesian product of the set X and Y.
It's given that none of the sets is empty.

Answer 1

We can construct a bijection between $(X \times Y) \times Z$ and $X \times (Y \times Z)$.

the bijection $f : (X \times Y) \times Z \to X \times (Y \times Z)$ can be defined as $f((x,y),z) = (x,(y,z))$

Now you can prove that this is a bijection by showing it has a two sided inverse.