Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?


share|cite|improve this question
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 '13 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 '13 at 11:26
AAA: see this definition! – MattAllegro Apr 24 '14 at 11:04
up vote 2 down vote accepted

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.