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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: $x_i \in \vec{x}=[x_1\;x_2\;\cdots\;x_n]^T\in \mathbb{R}^n$? And if I later want to specify 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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  • $\begingroup$ 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 $y_k=\pm1,k=1,\dots,n$". $\endgroup$
    – J. M. ain't a mathematician
    Dec 11, 2011 at 12:07
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    $\begingroup$ Vectors don't have elements; they have components, or entries, but not elements. So the use of the set membership symbol is not correct. $\endgroup$
    – Gerry Myerson
    Dec 11, 2011 at 12:12
  • $\begingroup$ @GerryMyerson mathworld.wolfram.com/RealVector.html This link seems suggest that a vector can have elements. $\endgroup$
    – High GPA
    Jun 7, 2022 at 3:44
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    $\begingroup$ @High, I've sent mathworld a message, asking them to consider changing their wording. $\endgroup$
    – Gerry Myerson
    Jun 7, 2022 at 5:21

1 Answer 1

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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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