What is the set represented by [n]

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:

1. What is the set represented by $[n]$? Is it the set of all of the natural numbers between 1 (or 0) and N?
2. What does $\epsilon = 0, 1$ mean? Does this mean that $\epsilon$ can be either 0 or 1? If so, shouldn't it be $\epsilon \in \{0,1\}$?

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.

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.

• I think the best notation would be $x \boldsymbol{\in} \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. This emphasizes the set of two values, a frequent point of confusion for students. Ah well, you can't really disagree with established convention.
– 6005
Feb 11 '18 at 0:09

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 $$\{0,1,2,\dots,n-1\}$$.

• I agree there could be alternating views and there is of course no "objective" answer, but by far the vast majority of texts I have encountered use $[n]=\{1,2,\dots,n\}$ (i.e. not including $0$). Is that different in your experience? Jan 31 '19 at 9:21