If I want $x_i$ to be an arbitrary element of a vector $\vec{x}$ can I use the following notation: $x_i \in \vec{x}=[x_1\;x_2\;\cdots\;x_n]^T\in R^n$ ? And if I later want to spesify the interval of each $x_i$ to be between 0 and 1, can I then say that $x_i \in [0,1]\;\forall i$ ? Is this mathematically correct usage of $\in$ for both cases?
The actual problem I have is that I want to say that $y_i\in\vec{y}$ for $i\in\{1,2,\cdots,n\}$ and that each $y_i$ is binary $y_i\in\{-1,+1\}$. Should I stick to something like $\vec{y}\in\{-1,+1\}^n$ instead?
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$\begingroup$This is not set theoretically correct, because $[x_1, ..., x_n]^T \neq \{x_1, ..., x_n\}$. Nevertheless, it is an accepted convention to refer to components of the vector $\vec y$ as $y_1, ..., y_n$. So you can use $y_1, ..., y_n$ without stating $y_i \in \vec y$. On a personal note, in your case I'll prefer $\vec y \in \{-1, 1\}^n$.
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