From: ram@zedat.fu-berlin.de   
      
   Peter Fairbrother  wrote or quoted:   
   >Hmmm, as we are talking about sets here, rather than individual points   
   >or values, does it matter whether a point in the target has more than   
   >one inverse?   
      
     It really does not make much difference. The proper term would be   
     "relation" instead of "function". A relation counts as "functional"   
     if every x matches up with at most one value y. And "function" is   
     basically just the usual shorthand for saying "functional relation".   
      
     A function "f", so that "f(1)=3" and "f(2)=3":   
      
   f   
   x y   
   1 3   
   2 3   
      
     . The inverse "r" has the value columns swapped:   
      
   r   
   x y   
   3 1   
   3 2   
      
     This "r" is a relation, but not a function.   
      
     Wikipedia: "The inverse of 'f' exists if and only if 'f'   
     is bijective". The "f" from above is not bijective (because   
     it's not injective), so the inverse of "f" does not exist.   
      
     One might define a function "g" using sets, "g(3)={1,2}":   
      
   g   
   x y   
   3 {1,2}   
      
     . But this is not the usual meaning of "inverse function", because   
     "f(g(x))" would be undefined instead of "x". However, "{1,2}" is the   
     preimage of "3". Conveniently, the "preimage" /is/ defined as a set:   
      
   |preimage - Wiktionary:   
   |   
   |preimage (plural preimages). (mathematics) For a given   
   |function, the /set/ of all elements of the domain that   
   |are mapped into a given subset of the codomain; ...   
      
     . So, the preimage might exist even if no inverse exists.   
      
     I only went over the standard terminology here, the kind you   
     pretty much get in school. Of course, anyone can come up   
     with their own version if they want. Some authors talk about   
     "multivalued functions", but that usually ends up being read   
     as "set-valued functions". The "full inverse" of a function   
     then gets defined as one of those "multivalued functions".   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)