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What does union mean in math sets?
What is union of sets with examples?
What does the U mean in math?
In a set theory development of the natural numbers, what we do is we start with nothing but sets and then build structures within that mimic our knowledge of numbers and operations on them. The standard construction for the natural numbers is to set
and then inductively define each following natural number by the successor function 
which gives the next number after 
as
So 
, 
, etc.
Note in this construction the only objects are the empty set and sets built from the empty set, the numbers are just names we assign to certain configurations of the sets. In this configuration, it is true that any natural number is just the set of all the previous numbers
Thus it makes the sense to take the union of the numbers because numbers are just names for special sets. And when you take the union of all of them, you get all natural numbers.
Added: For your exercise, you just want to show the two sets are equal, so as usual, show they are subsets of each other. So show any natural number is in that union, and then show that any member of that union is in fact a natural number
Second addendum: Technically the author should have written
but when the context of the index set is obvious, we often do abuse of notation and skip writing the index set and go straight to
The notation is not mysterious and is not “technically wrong” or “abusive” as others commented. You find it in several books; I learned it in the appendix of Kelley's book “General Topology”.
If 
is a set, then
For instance, if 
, then 
.
The book you link is not very consistent in notation, I should say.
In the “almost formal” set theory that's developed in the book, everything is a set. And the natural numbers are defined recursively by
and collecting what's obtained in this way in the set 
. Let's see what happens when the union is applied to 
instead of 
:
With 
we get
There seems to be a pattern. And indeed there is: if 
is a natural number and 
, then 
. Since, by definition,
you have that, when 
which you can prove by induction. It follows that
The notation 
is justified by a specific axiom of set theory: the axiom of union states that for any 
there exists 
such that, for every 
, there is 
with 
(probably not the same notation of your book). Using specification, we can isolate from that (possibly big) set the one that we need to use and denote it as shown above.
The “binary union” 
is defined to be 
.
Edit I thought amssymb provides \cupdot which does what you want... but it doesn't.
\usepackage{MnSymbol} provides \cupdot and \bigcupdot but is incompatible with amssymb which is unfortunate.
Sometimes disjoint union is depicted using \sqcup which has the advantage of being in amssymb
Another possibility to go around the problem that there is no such symbol in amssymb is to use the dot-accent: \dot{\bigcup} or also \dot\bigcup. This works for all symbols, and might very well be the reason that there are no dotted symbols in amssymb.
To let TeX treat such a new construct as an operator in terms of spacing though, you need to use \mathop and \mathbin, that's to say \mathop{\dot{\bigcup}} and \mathbin{\dot{\cup}}.
It is a union symbol. It is the union of all sets $S_i$, as $i$ ranges over all elements of the index set $I$.
$$\bigcup_{i\in I} S_i =\Bigl\{x\,\Bigm|\, \text{there exists }i\in I\text{ such that }x\in S_i\Bigr\}.$$
The symbol you describe is a special case, $$A_1\cup A_2 = \bigcup_{i\in\{1,2\}} A_i.$$
It is the union symbol. In this context, it is the union of all sets $S_i$, with $i\in I$.