# Formal definition of “ inverse operations” on a set?

Suppose f1 and f2 are two operations on A, that is functions from A² to A.

My question is : how to express in general and formally the fact that each operation is the inverse of the other?

Is it correct to say :

f1 and f2 are inverse operations on A iff for all a, b, c belonging to A

(1) if the image of (a,b) under f1 is c, then the image of (c,b) under f2 is a

AND

(2) if the image of (a,b) under f2 is c, then the image of (c,b) under f1 is a.

Remark. In the case of multiplication and division, maybe I would have to substract the set {0} and say " inverse operations on R - {0} "?

Is there an easier way to express the same fact using the concept of " inverse element" ? If it is the case, how to relate the " inverse element" definition and the general definiition of inverse operations?

• Yes, there is an easier way! “Division” is not really a separate operation from multiplication, it’s just multiplication by the inverse element. – Alex Sanger Apr 20 at 16:15

This may or may not helpful: I suppose you can say multiplication is a binary function $$\cdot:X^2\to X$$, and then say division is the partial binary function $$/:X^2\to X$$ defined by $$x/y=z$$ iff $$y\cdot z=x$$ AND $$y\cdot w\ne x$$ for all $$w\ne z\in X$$, else $$x/y$$ is undefined.
For example: Let $$X=\mathbb{N}=\{0,1,\ldots\}$$. Then $$6/2=3$$ since $$2\cdot 3=6$$ and $$2\cdot w\ne 6$$ for all $$w\ne 3\in\mathbb{N}$$.
For all $$n\in\mathbb{N}$$, $$n/0$$ is undefined: If $$n\ne0$$, then $$0\cdot z=0\ne n$$ for all $$z\in\mathbb{N}$$. If $$n=0$$, then $$0\cdot 0=0$$ but $$0\cdot 1=0$$, so $$n/0$$ is undefined. I'll leave you to prove that for all $$n\ne0\in\mathbb{N}$$, $$0/n=0$$.