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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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For your "actual problem": Usually, when you have $\vec{y}$ (or the notation I'm more accustomed to, $\mathbf y$) and $y_k$ in close proximity, people will quickly get that you intend the former to be a vector and the latter to be a component of said vector (and similarly for matrices). I'd just say something along the lines of "$n$-vector $\vec{y}$, with $y_k=\pm1,k=1,\dots,n$". –  J. M. Dec 11 '11 at 12:07
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Vectors don't have elements; they have components, or entries, but not elements. So the use of the set membership symbol is not correct. –  Gerry Myerson Dec 11 '11 at 12:12
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up vote 1 down vote accepted

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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