Functional notation usage for arbitrary relation? Question about questionable notation used in a homework assignment.
The assignment defines a relation f := { (0,1), (1,3), (2,1) }. It then asks to "show" this is a function. IMO strictly (i.e. Bourbaki) speaking it is not but so be it. The intention is clear.
It further asks to calculate f[2], f(2), f-1[1], f-1(1), ...
It is understood that the author uses Kuratowski's definition of ordered pairs, the natural numbers as n = {0, ..., n-1} and that f-1 is the inverse relation
f-1 = { (1,0), (3,1), (1,2) }, which is obviously not a function.
So my question is: what could f-1(.) possibly mean for the non-functional relation f-1? It is sometimes used as a sloppy alternative notation for f-1[.] but it is obvious from the question that the author makes a strict distinction between [.] and (.). And that whatever (.) is supposed to mean, its definition should generalize Euler's notation (.) for functions.

The inverse of a function f is f$^{-1}$ when the inverse of f exists.
The inverse of a relation R is R$^{-1}$ = { (x,y) : yRx }.
The set extensions of a function f are f[A] = { f(x) : x in A }
and f$^{-1}$[A] = { x : f(x) in A }.  
