Are they considered equal in some sense?
For instance, I always write "...for vector $\mathbf{x} \in {\mathbb{R}}^n$ we have ...". I have a small problem with this (not a big one). The problem comes from the fact that the "cartesian power" of a set is defined as: ${\mathbb{R}}^n = \underbrace{ \mathbb{R} \times \mathbb{R} \times \cdots \times \mathbb{R} }_{n}= \{ (x_1,\ldots,x_n) \mid x_i \in \mathbb{R} \ \text{for all} \ 1 \le i \le n \}$. In other words, we are implying that the vector $\mathbf{x}$ is a tuple because we, with the relation "is an element of" ($\in$), are defining the $n$-dimensional vector to be a member of a set ${\mathbb{R}}^n$ in where members are $n$-tuples.
Is there implied a tiny mathematical abuse of notation here? Or have I completely lost it? :)