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.

@JasonDeVito , I believe that means equivalent relations. I'm not sure how to go about it. — Just another person, Sep 25 '20 at 14:51
There is a canonical bijection between each of these three expressions and they are often used interchangeably, however they are not technically the same. Consider the simple case of $X,Y,Z$ all being the one element sets ${x},{y},{z}$ respectively. $X\times Y\times Z$ would result in ${(x,y,z)}$. $X\times (Y\times Z)$ would result in ${(x,(y,z))}$ and $(X\times Y)\times Z$ would result in ${((x,y),z)}$ — JMoravitz, Sep 25 '20 at 14:55
See this discussion about the so-called associativity of cartesian product — Jean Marie, Sep 25 '20 at 14:57

score 0 · Answer 1 · answered Sep 25 '20 at 14:54

0

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.

answered Sep 25 '20 at 14:54

That we can construct such a bijection does not directly answer the question of "are these sets equal?" – JMoravitz Sep 25 '20 at 14:56

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