# Notation for elements of a vector

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?

• 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 $y_k=\pm1,k=1,\dots,n$". Dec 11, 2011 at 12:07
• Vectors don't have elements; they have components, or entries, but not elements. So the use of the set membership symbol is not correct. Dec 11, 2011 at 12:12
• @GerryMyerson mathworld.wolfram.com/RealVector.html This link seems suggest that a vector can have elements. Jun 7, 2022 at 3:44
• @High, I've sent mathworld a message, asking them to consider changing their wording. Jun 7, 2022 at 5:21

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 $\vec y \in \{-1, 1\}^n$.