nLab
range

Classically, the range of a function ff with domain AA is the set {f(x)xA}\{f(x) \;|\; x \in A\} (whose existence, in material set theory, is given by the axiom of replacement). As we came to realise that a function should be given with a codomain (which is automatic in structural set theory), the term ‘range’ generalised in two ways:

  • as the codomain itself, so that the earlier terminology is then preserved only for surjections;
  • as the image
    {y:Bx:A,y=f(x)} \{y \colon B \;|\; \exists x\colon A,\; y = f(x)\}

    (whose existence, in axiomatic set theory, is given by the much weaker axiom of bounded separation) of f:ABf\colon A \to B.

The former generalisation was historically common (and is sometimes still used) in groupoid theory; the latter is what we usually mean today.

Note that the axiom of replacement is still needed for a function (such as a family of sets) whose codomain is a proper class, to prove that its image is small when its domain is small.

Revised on September 5, 2011 16:11:23 by Toby Bartels (75.88.82.16)