What is the meaning of this set notation? I am studying discrete mathematics using Rosen Discrete Mathematics 7th Edition. I am doing sets. I don't understand what this means.

I don't understand why the intersection of all these sets is {1}. I thought it would be all {1,2,3,...., i}.
 A: Remember that if $P\subset Q$, then $P \cap Q  = P$. it means that in intersection the common inner parts are the result, since $P$ is all inside $Q$, then the result must be $P$. The same logic is for your example sets:
$A_1 = \{1\} \subset \underbrace{\{1,2\}}_{A_2}\subset\underbrace{\{1,2,3\}}_{A_3}\subset \cdots\subset A_n,$
So, for all $n\ge1$, we have that $A_1 = \{1\}\subseteq A_n$. Thus, $\displaystyle\bigcap_{n\in\mathbb{N}}A_n = A_1 = \{1\}$
(Just think : what elements are always inside all $A_n$ ?
A: Note that an element $a$ belongs to the intersection of two sets $A_i \cap A_j$ if and only if $a\in A_i$ and $a\in A_j$. The only such element given the infinite sequence of sets you're presented with is $1$ - precisely because $A_1 = \{1\}$ is the smallest (in terms of cardinality) set in the sequence, and is contained as a subset in all the others.
If you find the $\{1,2,3,\ldots\}$ for $i=1,2,3,\ldots$ notation confusing, this is just mathematial convention and the ellipsis represents an infinite continuation of elements. Note, too, that $1,2,3$ aren't always in the set, which is easy to see by plugging in $i=1$, for example.
A: Think it recursively.
When $i = 1, 2, 3, \dots$ you get the sets $\{1\}, \{1,2\}, \{1,2,3\}, \dots$
Observe that $\{1\}\cap \{1,2\} = \{1\}$, then $\left(\{1\}\cap \{1,2\}\right) \cap \{1,2,3\} = \{1\} \cap \{1,2,3\} = \{1\}$. Recursively, you have the answer.
