# 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 2/3 $\not\in$ $\mathbb{N}_{>0}$

So how is division a binary operation?

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Of course you are right. That page is a bit sloppy in speaking about addition, subtraction, multiplication and division on an arbitrary nonempty set. – Andreas Caranti Feb 9 at 10:45
Division is a binary operation on, say, the positive rationals, or the positive reals. But as you note it's not a binary operation on the positive integers. – Gerry Myerson Feb 9 at 11:26

## 1 Answer

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.

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