What is Binary Operation -- is division a binary operation? I was reading the definition of a binary operation of here.
The thing I don't understand is how is division a binary operation?
If you consider division with pairs in $\mathbb{N}_{>0}\times\mathbb{N}_{>0}$, you do not neccesary get an elenment in $\mathbb{N}_{>0}$, e.g. $(2,3)\in\mathbb{N}_{>0}\times\mathbb{N}_{>0}$ but $\frac{2}{3}\not\in\mathbb{N}_{>0}$.
So how is division a binary operation?
 A: As the definition demonstrates, you can only talk about a binary operation on a given set $A$. To say any given operation is a binary operation, you need to specify what the set $A$ is. For your example, division is a binary operation on $\mathbb{Q}\setminus\{0\}$ for example (it is also a binary operation on $\mathbb{R}\setminus\{0\}$), but it is not a binary operation on $\mathbb{N}_{> 0}$, as you point out. 
As Andreas Caranti mentions in his comment, the following sentence (found on the linked page) is a bit sloppy.

"Examples of binary operation on $A$ from $A\times A$ to $A$ include addition $(+)$, subtraction $(-)$, multiplication $(\times)$ and division $(\div)$."

They probably should have said something along the lines of:

Addition $(+)$, subtraction $(-)$, multiplication $(\times)$ and division $(\div)$ are examples of binary operations (for the appropriate choice of set $A$ in each case).


A binary operation on a non-empty set $A$ is a function $f : A\times A \to A$, so technically the set $A$ is specified implicitly by $f$; however, the words addition, subtraction, multiplication, and division do not implicitly specify a particular set.
