Trying to get it right regarding " inverse" applied to relations, functions and operations...

Question

[ The discussion ( see below) tends to show that my question was based on the erroneous assumption that the expression " inverse operation" is standard. Apparently, it is an informal expression used in the context of elementary arithmetic, when, for example, one says that multiplication and addition are inverse operations].

The term " inverse" can qualify different sorts of things, and its meaning is not exactly the same in all cases.

I'd like to make sure I understand things corrrectly.

Are the following propositions all correct? ( I mark with ??? the most dubious to me. Some of the propositions are simple conjectures. )

(1) Every relation ( set of ordered pairs) has an inverse relation.

(2) The inverse relation of a function is not necessarily a function. ( One cannot say , as a general rule : " since g is the inverse relation of the function f, g is f's inverse function" ).

Remark . The symbol "f^-1" should not be read systematically as " the inverse function of f". It only means " the inverse relation of f". ( Is this right?)

(3) The inverse relation of a function is its " inverse function" just in case this inverse relation is itself a function.

(4) A function f has an inverse function iff f is (a) injective and (b) surjective.

(5) The inverse relation of a binary operation on A ( function from A² to A) is never a binary operation on A.

(6) The inverse relation of an operation on A is possibly a function from A to A².

Remark. But are there cases in which such a function would be interesting mathematically ( function from a set A to the cartesain product A²)?

(7) If f1 is an operation on A ( function from A² to A) then its inverse operation ( if it exists) - let us call it f2 - has nothing to do with the inverse relation of f1. ( " Inverse operation " is a totally different concept from " inverse function" and from " inverse relation") ( ???)

(8) The inverse operation of an operation is ( informally )the operation that " undoes" what the first has done. ( For example: division is the inverse operation of multiplication since it " undoes" what multiplication has done. 2.7=14 and 14/7 = 2 ).

(9) The inverse operation can be defined formally as follows :

f2 is the inverse of f1 iff, for all a, b, c belonging to A :

          **( (a,b), c ) belongs to f1   <--> ( (c,b), a ) belongs to f2.** 

(???)

(10) It is possible that an operation on A has no inverse operation on A. ( The existence of a binary operation on A does not imply the existence of an inverse binary operation). ( ???)

(11) The " inverse operation" relation is symmetric : if operation f2 is the inverse of operation f1, then f1 is the inverse operation of f2.

(12) The " inverse operation " relation is " exclusive": no operation may have more than one inverse operation. ( ???)

(13) The existence of an inverse binary operation on A requires the existence of an inverse element for every element of A ( except maybe for some exceptionary element). (???)

Answer 1

1. True
2. True
3. True
4. True, but... say $f:X \to Y$ is a function. If you want an inverse function $g:Y \to X$, then yes, $f$ must be injective and surjective. However, if $f$ is not surjective (so $f(X)\neq Y$), then you can still define an inverse function $g:f(X) \to X$. For instance, $\arctan(x):\mathbb{R} \to \mathbb{R}$ is injective but not surjective. However, if we consider it as a function $\mathbb{R}\to (-\pi/2,\pi/2)$, then it does have an inverse function, namely $\tan(x):(-\pi/2,\pi/2) \to \mathbb{R}$. We need to restrict the domain of $\tan(x)$ for them to truly be inverse functions though, because for instance $\tan(2\pi)= 0$ while $\arctan(0) \neq 2\pi$.
5. True. If you are considering a function $f:A \times A \to A$ as $\{((a,b),c) \mid f(a,b)=c\} \subseteq (A \times A) \times A$, then the inverse relation will be a subset of $A \times (A \times A)$, so not a function on $A\times A$.
6. True. And sure, a function $A \to A^2$ could be interesting mathematically. A function $\mathbb{N} \to \mathbb{N}^2$ shows that $\mathbb{N}^2$ is countable, for instance.

For the rest of these, you need to say explicitly what you mean by "inverse operation". We can't say what it is "informally" or what properties it has if we don't know what it is formally.