$\{1,\dots,n\}$ - Without further definitions, does that imply $n \geq 1$? I have another extremely basic notation question:
In many books, wikipedia and other places they write $\{1,\dots,n\}$ without ever defining n.
Does that imply that $ n \in \mathbb{N} $ or does that also allow $ n \in \mathbb{Z} $, which would mean that the set could also be empty?
(Sorry if this has already been answered somewhere, I find it very hard to find the right search terms for these kind of questions)
 A: Usually, when we have a set of the form
$$\{1, \dots, n\}$$
we are referring to the set of natural numbers from $1$ to $n,$ where $n$ is some natural number ($n \in \mathbb{N}$).
Also, note that $\emptyset$ is not “included” in the above set, for some object $x$ be included in a set $X$ it has to be the case that $x \in X,$ i.e., $x$ is an element of $X.$
In this case, for $n \in \mathbb{N}$ it is clear that $\emptyset \notin \{1,\dots, n\}.$
Although, the empty set is a subset of $\{1,\dots,n\},$ i.e., $\emptyset \subseteq \{1,\dots,n\}$ (it is the case that $\emptyset$ is a subset of any set).
A: It depends on the context and the intentions of the author. In many
instances we define $\,[n] := \{1,2,\dots,n\}\,$ and intend it to be
used when $\,n>0.\,$ However, in other instances, we allow $\,n=0\,$
in which case $\,[0] = \emptyset := \{\} \,$ is the set with zero elements. It
all depends on the situation and the author. Sometimes $\,n=0\,$ does
makes sense, but at other times it does not. You can think of
this as merely a matter of convention exactly similar to the
convention that an empty summation is equal to zero.
Note that $\,[m]\subseteq [n]\,$ iff $\,m\le n\,$ which includes
the special case $\,\emptyset = [0]\subseteq [n]\,$ for all
$\,n\ge 0.\,$
A: Well, if you consider the set $\{1,\ldots,n\}$ where $n\geq 0$ is an integer, then the case $n=0$ is defined as the empty set.
