While reading the paper "On Span Programs" by Karchmer and Wigderson (1993) I have come across the following definition:
The highlighted part of the definition contains the following elements that I find confusing:
On
In this context, $[n] := \{1,2,\dots,n\}$.
Here, $\epsilon = 0, 1$ indicates $\epsilon \in \{0,1\}$, but it's been abbreviated to avoid extra curly braces.
By analogy, $x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$ indicates $x \in \{\frac{-b + \sqrt{b^2-4ac}}{2a},\frac{-b - \sqrt{b^2-4ac}}{2a}\}$, but it's much neater to write it the first way.
On
There are alternating views as to whether $[n]$ should refer to $\{1,2,3,\dots,n\}$ or whether $[n]$ should refer to $\{0,1,2,\dots,n-1\}$, similarly to how some people prefer that $\Bbb N$ includes zero while others prefer it does not include zero.
In either case, $[n]$ is the prototypical $n$-element set using the first consecutive $n$ natural numbers. In most scenarios it does not even matter which of the two sets are actually being intended since the point is for the set to be a set with $n$ elements and it doesn't actually matter what those $n$ elements really are and there is an obvious bijection between $\{1,2,3,\dots,n\}$ and $\{0,1,2,\dots,n-1\}$.
Yes, for both questions. We have $[n] = \mathbb Z \cap [1,n]$. You’re right that $\epsilon \in \{0,1\}$ is probably better way to write this, but maybe the authors thought it looked clumsy since the notation was already inside a set.