What is the difference between these two simple sets? What is the difference between these two sets:
$\mathbb{R}^{d}\backslash\mathbb{Q}^{d}$   and    $\left(\mathbb{R}\backslash\mathbb{Q}\right)^{d}$
???
 A: Generally, we have
$$(A\times B) \setminus (C\times D) = (A\setminus C) \times B \cup A \times (B\setminus D),$$
which is equal to $(A\setminus C)\times (B\setminus D)$ if and only if


*

*$A = \varnothing$, or

*$B = \varnothing$, or

*$A\setminus C = \varnothing = B\setminus D$, or

*$A\setminus C = A$ and $B\setminus D = B$.


The principle remains the same for more factors.
In this example, $\mathbb{R}^d\setminus \mathbb{Q}^d$ is the set of points with at least one irrational coordinate, and $(\mathbb{R}\setminus\mathbb{Q})^d$ is the set of points with all coordinates irrational. The two sets are equal if and only if $d \leqslant 1$.
A: The point $(1,1,\sqrt2)$ is contained in $\mathbb R^3\setminus\mathbb Q^3$, but it is not contained in $(\mathbb R\setminus\mathbb Q)^d$.  
$\mathbb R^d$ - tuples of points in $\mathbb R$.
$\mathbb Q^d$ - tuples of points in $\mathbb Q$.  
$\mathbb R^d\setminus\mathbb Q^d$ - tuples of points in $\mathbb R$ that are not tuples of points in $\mathbb Q$; i.e., not every coordinate is a member of $\mathbb Q$.  
On the other hand...
$\mathbb R\setminus\mathbb Q$ - irrational numbers (members of $\mathbb R$ that are not members of $\mathbb Q$).
$(\mathbb R\setminus\mathbb Q)^d$ - tuples of irrational numbers.  
$(\mathbb R\setminus\mathbb Q)^d$ is the smaller set: in order to be a member of that, you need to have every coordinate not in $\mathbb Q$.  $\mathbb R^d\setminus\mathbb Q^d$ is larger: to be in that set, you only need to have one coordinate not in $\mathbb Q$.  
