Is it true that $\mathbb{R}^m \cap \mathbb{R}^n = \emptyset$ where $m \neq n$ Is it true that $\mathbb{R}^m \cap \mathbb{R}^n = \emptyset$ where $m \neq n$ ?
I would say that it is true because the elements in each set are different entities. This means that one set cannot be a subset of the other. An example would be $\mathbb{R} \cap \mathbb{R}^2$. You cannot say that $1 \in \mathbb{R}^2$ for example.

Well, these sets are disjoint.
To see this, define the ordered pair
$$(x,y) = \{\{x\},\{x,y\}\}$$
and extend this definition to $n$-tuples
$$(x_1,\ldots,x_n) = ((x_1,\ldots,x_{n-1}),x_n).$$
So $n$-tuples are sets and we have clear notion of equality of two sets:
$$A= B :\Longleftrightarrow \forall x[x\in A\Leftrightarrow x\in B].$$
Equality of two $n$-tuples is then given by
$$(x_1,\ldots,x_n)=(y_1,\ldots,y_n)\Leftrightarrow \forall i[x_i=y_i].$$
Taking this into accout you see that the above sets are disjoint if $m\ne n$.
